2015
2015
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Paper 1, Section I,
2015 commentFind the following limits: (a) (b) (c)
Carefully justify your answers.
[You may use standard results provided that they are clearly stated.]
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Paper 1, Section I, E
2015 commentLet be a complex power series. State carefully what it means for the power series to have radius of convergence , with .
Find the radius of convergence of , where is a fixed polynomial in with coefficients in .
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Paper 1, Section II, F
2015 commentLet be sequences of real numbers. Let and set . Show that for any we have
Suppose that the series converges and that is bounded and monotonic. Does converge?
Assume again that converges. Does converge?
Justify your answers.
[You may use the fact that a sequence of real numbers converges if and only if it is a Cauchy sequence.]
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Paper 1, Section II, 10D
2015 comment(a) For real numbers such that , let be a continuous function. Prove that is bounded on , and that attains its supremum and infimum on .
(b) For , define
Find the set of points at which is continuous.
Does attain its supremum on
Does attain its supremum on ?
Justify your answers.
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Paper 1, Section II,
2015 comment(i) State and prove the intermediate value theorem.
(ii) Let be a continuous function. The chord joining the points and of the curve is said to be horizontal if . Suppose that the chord joining the points and is horizontal. By considering the function defined on by
or otherwise, show that the curve has a horizontal chord of length in . Show, more generally, that it has a horizontal chord of length for each positive integer .
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Paper 1, Section II, E
2015 commentLet be a bounded function, and let denote the dissection of . Prove that is Riemann integrable if and only if the difference between the upper and lower sums of with respect to the dissection tends to zero as tends to infinity.
Suppose that is Riemann integrable and is continuously differentiable. Prove that is Riemann integrable.
[You may use the mean value theorem provided that it is clearly stated.]
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Paper 2, Section I, B
2015 commentFind the general solution of the equation
where is a constant not equal to 2 .
By subtracting from the particular integral an appropriate multiple of the complementary function, obtain the limit as of the general solution of and confirm that it yields the general solution for .
Solve equation with and .
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Paper 2, Section , B
2015 commentFind the general solution of the equation
Compute all possible limiting values of as .
Find a non-zero value of such that for all .
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Paper 2, Section II, B
2015 commentWrite as a system of two first-order equations the second-order equation
where is a small, positive constant, and find its equilibrium points. What is the nature of these points?
Draw the trajectories in the plane, where , in the neighbourhood of two typical equilibrium points.
By considering the cases of and separately, find explicit expressions for as a function of . Discuss how the second term in affects the nature of the equilibrium points.
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Paper 2, Section II, B
2015 commentSuppose that satisfies the equation
where is a given non-zero function. Show that under the change of coordinates ,
where a dot denotes differentiation with respect to . Furthermore, show that the function
satisfies
Choosing , deduce that
for some appropriate function . Assuming that may be neglected, deduce that can be approximated by
where are constants and are functions that you should determine in terms of .
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Paper 2, Section II, B
2015 commentSuppose that obeys the differential equation
where is a constant real matrix.
(i) Suppose that has distinct eigenvalues with corresponding eigenvectors . Explain why may be expressed in the form and deduce by substitution that the general solution of is
where are constants.
(ii) What is the general solution of if , but there are still three linearly independent eigenvectors?
(iii) Suppose again that , but now there are only two linearly independent eigenvectors: corresponding to and corresponding to . Suppose that a vector satisfying the equation exists, where denotes the identity matrix. Show that is linearly independent of and , and hence or otherwise find the general solution of .
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Paper 4, Section I, C
2015 commentFind the moment of inertia of a uniform sphere of mass and radius about an axis through its centre.
The kinetic energy of any rigid body with total mass , centre of mass , moment of inertia about an axis of rotation through , and angular velocity about that same axis, is given by . What physical interpretation can be given to the two parts of this expression?
A spherical marble of uniform density and mass rolls without slipping at speed along a flat surface. Explaining any relationship that you use between its speed and angular velocity, show that the kinetic energy of the marble is .
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Paper 2, Section II, B
2015 commentConsider the equation
for the function , where and are real variables. By using the change of variables
where and are appropriately chosen integers, transform into the equation
Hence, solve equation supplemented with the boundary conditions
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Paper 4, Section I, C
2015 commentWrite down the 4-momentum of a particle with energy and 3-momentum p. State the relationship between the energy and wavelength of a photon.
An electron of mass is at rest at the origin of the laboratory frame: write down its 4 -momentum. The electron is scattered by a photon of wavelength travelling along the -axis: write down the initial 4-momentum of the photon. Afterwards, the photon has wavelength and has been deflected through an angle . Show that
where is the speed of light and is Planck's constant.
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Paper 4, Section II,
2015 commentA particle is projected vertically upwards at speed from the surface of the Earth, which may be treated as a perfect sphere. The variation of gravity with height should not be ignored, but the rotation of the Earth should be. Show that the height of the particle obeys
where is the radius of the Earth and is the acceleration due to gravity measured at the Earth's surface.
Using dimensional analysis, show that the maximum height of the particle and the time taken to reach that height are given by
where and are functions of .
Write down the equation of conservation of energy and deduce that
Hence or otherwise show that
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Paper 4, Section II, C
2015 commentA particle of mass and charge has position vector and moves in a constant, uniform magnetic field so that its equation of motion is
Let be the particle's angular momentum. Show that
is a constant of the motion. Explain why the kinetic energy is also constant, and show that it may be written in the form
where and .
[Hint: Consider u
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Paper 4, Section II, C
2015 commentConsider a particle with position vector moving in a plane described by polar coordinates . Obtain expressions for the radial and transverse components of the velocity and acceleration .
A charged particle of unit mass moves in the electric field of another charge that is fixed at the origin. The electrostatic force on the particle is in the radial direction, where is a positive constant. The motion takes place in an unusual medium that resists radial motion but not tangential motion, so there is an additional radial force where is a positive constant. Show that the particle's motion lies in a plane. Using polar coordinates in that plane, show also that its angular momentum is constant.
Obtain the equation of motion
where , and find its general solution assuming that . Show that so long as the motion remains bounded it eventually becomes circular with radius .
Obtain the expression
for the particle's total energy, that is, its kinetic energy plus its electrostatic potential energy. Hence, or otherwise, show that the energy is a decreasing function of time.
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Paper 4, Section II, C
2015 commentWrite down the Lorentz transform relating the components of a 4-vector between two inertial frames.
A particle moves along the -axis of an inertial frame. Its position at time is , its velocity is , and its 4 -position is , where is the speed of light. The particle's 4-velocity is given by and its 4 -acceleration is , where proper time is defined by . Show that
where and .
The proper 3-acceleration a of the particle is defined to be the spatial component of its 4-acceleration measured in the particle's instantaneous rest frame. By transforming to the rest frame, or otherwise, show that
Given that the particle moves with constant proper 3 -acceleration starting from rest at the origin, show that
and that, if , then .
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Paper 3, Section I, D
2015 commentSay that a group is dihedral if it has two generators and , such that has order (greater than or equal to 2 and possibly infinite), has order 2 , and . In particular the groups and are regarded as dihedral groups. Prove that:
(i) any dihedral group can be generated by two elements of order 2 ;
(ii) any group generated by two elements of order 2 is dihedral; and
(iii) any non-trivial quotient group of a dihedral group is dihedral.
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Paper 3, Section I, D
2015 commentHow many cyclic subgroups (including the trivial subgroup) does contain? Exhibit two isomorphic subgroups of which are not conjugate.
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Paper 3, Section II, D
2015 commentWhat does it mean for a group to act on a set ? For , what is meant by the orbit to which belongs, and by the stabiliser of ? Show that is a subgroup of . Prove that, if is finite, then .
(a) Prove that the symmetric group acts on the set of all polynomials in variables , if we define to be the polynomial given by
for and . Find the orbit of under . Find also the order of the stabiliser of .
(b) Let be fixed positive integers such that . Let be the set of all subsets of size of the set . Show that acts on by defining to be the set , for any and . Prove that is transitive in its action on . Find also the size of the stabiliser of .
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Paper 3, Section II, D
2015 commentLet be groups and let be a function. What does it mean to say that is a homomorphism with kernel ? Show that if has order 2 then for each . [If you use any general results about kernels of homomorphisms, then you should prove them.]
Which of the following four statements are true, and which are false? Justify your answers.
(a) There is a homomorphism from the orthogonal group to a group of order 2 with kernel the special orthogonal group .
(b) There is a homomorphism from the symmetry group of an equilateral triangle to a group of order 2 with kernel of order 3 .
(c) There is a homomorphism from to with kernel of order 2 .
(d) There is a homomorphism from to a group of order 3 with kernel of order 2 .
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Paper 3, Section II, D
2015 comment(a) State and prove Lagrange's theorem.
(b) Let be a group and let be fixed subgroups of . For each , any set of the form is called an double coset, or simply a double coset if and are understood. Prove that every element of lies in some double coset, and that any two double cosets either coincide or are disjoint.
Let be a finite group. Which of the following three statements are true, and which are false? Justify your answers.
(i) The size of a double coset divides the order of .
(ii) Different double cosets for the same pair of subgroups have the same size.
(iii) The number of double cosets divides the order of .
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Paper 3, Section II, D
2015 comment(a) Let be a non-trivial group and let for all . Show that is a normal subgroup of . If the order of is a power of a prime, show that is non-trivial.
(b) The Heisenberg group is the set of all matrices of the form
with . Show that is a subgroup of the group of non-singular real matrices under matrix multiplication.
Find and show that is isomorphic to under vector addition.
(c) For prime, the modular Heisenberg group is defined as in (b), except that and now lie in the field of elements. Write down . Find both and in terms of generators and relations.
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Paper 4, Section I, E
2015 comment(a) Find all integers and such that
(b) Show that if an integer is composite then .
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Paper 4 , Section I, E
2015 commentState the Chinese remainder theorem and Fermat's theorem. Prove that
for any prime .
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Paper 4, Section II, E
2015 comment(i) Let be an equivalence relation on a set . What is an equivalence class of ? What is a partition of Prove that the equivalence classes of form a partition of .
(ii) Let be the relation on the natural numbers defined by
Show that is an equivalence relation, and show that it has infinitely many equivalence classes, all but one of which are infinite.
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Paper 4, Section II, E
2015 commentLet be a prime. A base expansion of an integer is an expression
for some natural number , with for .
(i) Show that the sequence of coefficients appearing in a base expansion of is unique, up to extending the sequence by zeroes.
(ii) Show that
and hence, by considering the polynomial or otherwise, deduce that
(iii) If is a base expansion of , then, by considering the polynomial or otherwise, show that
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Paper 4, Section II, E
2015 commentState the inclusion-exclusion principle.
Let . A permutation of the set is said to contain a transposition if there exist with such that and . Derive a formula for the number, , of permutations which do not contain a transposition, and show that
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Paper 4, Section II, E
2015 commentWhat does it mean for a set to be countable? Prove that
(a) if is countable and is injective, then is countable;
(b) if is countable and is surjective, then is countable.
Prove that is countable, and deduce that
(i) if and are countable, then so is ;
(ii) is countable.
Let be a collection of circles in the plane such that for each point on the -axis, there is a circle in passing through the point which has the -axis tangent to the circle at . Show that contains a pair of circles that intersect.
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Paper 2, Section I, F
2015 commentLet be a uniform random variable on , and let .
(a) Find the distribution of the random variable .
(b) Define a new random variable as follows: suppose a fair coin is tossed, and if it lands heads we set whereas if it lands tails we set . Find the probability density function of .
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Paper 2, Section I, F
2015 commentLet be events in the sample space such that and . The event is said to attract if the conditional probability is greater than , otherwise it is said that repels . Show that if attracts , then attracts . Does repel
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Paper 2, Section II, F
2015 commentLionel and Cristiana have and million pounds, respectively, where . They play a series of independent football games in each of which the winner receives one million pounds from the loser (a draw cannot occur). They stop when one player has lost his or her entire fortune. Lionel wins each game with probability and Cristiana wins with probability , where . Find the expected number of games before they stop playing.
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Paper 2, Section II, F
2015 commentConsider the function
Show that defines a probability density function. If a random variable has probability density function , find the moment generating function of , and find all moments , .
Now define
Show that for every ,
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Paper 2, Section II, F
2015 commentState and prove Markov's inequality and Chebyshev's inequality, and deduce the weak law of large numbers.
If is a random variable with mean zero and finite variance , prove that for any ,
[Hint: Show first that for every .]
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Paper 2, Section II, F
2015 commentWhen coin is tossed it comes up heads with probability , whereas coin comes up heads with probability . Suppose one of these coins is randomly chosen and is tossed twice. If both tosses come up heads, what is the probability that coin was tossed? Justify your answer.
In each draw of a lottery, an integer is picked independently at random from the first integers , with replacement. What is the probability that in a sample of successive draws the numbers are drawn in a non-decreasing sequence? Justify your answer.
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Paper 3, Section I, A
2015 comment(i) For with , show that
(ii) Consider the vector fields and , where is a constant vector in and is the unit vector in the direction of . Using suffix notation, or otherwise, find the divergence and the curl of each of and .
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Paper 3, Section I, A
2015 commentThe smooth curve in is given in parametrised form by the function . Let denote arc length measured along the curve.
(a) Express the tangent in terms of the derivative , and show that .
(b) Find an expression for in terms of derivatives of with respect to , and show that the curvature is given by
[Hint: You may find the identity helpful.]
(c) For the curve
with , find the curvature as a function of .
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Paper 3, Section II, A
2015 commentThe vector field is given in terms of cylindrical polar coordinates by
where is a differentiable function of , and is the unit basis vector with respect to the coordinate . Compute the partial derivatives , and hence find the divergence in terms of and .
The domain is bounded by the surface , by the cylinder , and by the planes and . Sketch and compute its volume.
Find the most general function such that , and verify the divergence theorem for the corresponding vector field in .
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Paper 3, Section II, A
2015 commentState Stokes' theorem.
Let be the surface in given by , where and is a positive constant. Sketch the surface for representative values of and find the surface element with respect to the Cartesian coordinates and .
Compute for the vector field
and verify Stokes' theorem for on the surface for every value of .
Now compute for the vector field
and find the line integral for the boundary of the surface . Is it possible to obtain this result using Stokes' theorem? Justify your answer.
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Paper 3, Section II, A
2015 comment(i) Starting with the divergence theorem, derive Green's first theorem
(ii) The function satisfies Laplace's equation in the volume with given boundary conditions for all . Show that is the only such function. Deduce that if is constant on then it is constant in the whole volume .
(iii) Suppose that satisfies Laplace's equation in the volume . Let be the sphere of radius centred at the origin and contained in . The function is defined by
By considering the derivative , and by introducing the Jacobian in spherical polar coordinates and using the divergence theorem, or otherwise, show that is constant and that .
(iv) Let denote the maximum of on and the minimum of on . By using the result from (iii), or otherwise, show that .
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Paper 3, Section II, A
2015 comment(a) Let be a rank 2 tensor whose components are invariant under rotations through an angle about each of the three coordinate axes. Show that is diagonal.
(b) An array of numbers is given in one orthonormal basis as and in another rotated basis as . By using the invariance of the determinant of any rank 2 tensor, or otherwise, prove that is not a tensor.
(c) Let be an array of numbers and a tensor. Determine whether the following statements are true or false. Justify your answers.
(i) If is a scalar for any rank 2 tensor , then is a rank 2 tensor.
(ii) If is a scalar for any symmetric rank 2 tensor , then is a rank 2 tensor.
(iii) If is antisymmetric and is a scalar for any symmetric rank 2 tensor , then is an antisymmetric rank 2 tensor.
(iv) If is antisymmetric and is a scalar for any antisymmetric rank 2 tensor , then is an antisymmetric rank 2 tensor.
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Paper 1, Section I, B
2015 comment(a) Describe geometrically the curve
where and are positive, distinct, real constants.
(b) Let be a real number not equal to an integer multiple of . Show that
and derive a similar expression for .
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Paper 1, Section I,
2015 commentPrecisely one of the four matrices specified below is not orthogonal. Which is it?
Give a brief justification.
Given that the four matrices represent transformations of corresponding (in no particular order) to a rotation, a reflection, a combination of a rotation and a reflection, and none of these, identify each matrix. Explain your reasoning.
[Hint: For two of the matrices, and say, you may find it helpful to calculate and , where is the identity matrix.]
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Paper 1, Section II, 5B
2015 comment(i) State and prove the Cauchy-Schwarz inequality for vectors in . Deduce the inequalities
for .
(ii) Show that every point on the intersection of the planes
where , satisfies
What happens if
(iii) Using your results from part (i), or otherwise, show that for any ,
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Paper 1, Section II,
2015 comment(i) Consider the map from to represented by the matrix
where . Find the image and kernel of the map for each value of .
(ii) Show that any linear map may be written in the form for some fixed vector . Show further that is uniquely determined by .
It is given that and that the vectors
lie in the kernel of . Determine the set of possible values of a.
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Paper 1, Section II, A
2015 comment(i) Find the eigenvalues and eigenvectors of the following matrices and show that both are diagonalisable:
(ii) Show that, if two real matrices can both be diagonalised using the same basis transformation, then they commute.
(iii) Suppose now that two real matrices and commute and that has distinct eigenvalues. Show that for any eigenvector of the vector is a scalar multiple of . Deduce that there exists a common basis transformation that diagonalises both matrices.
(iv) Show that and satisfy the conditions in (iii) and find a matrix such that both of the matrices and are diagonal.
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Paper 1, Section II, A
2015 comment(a) A matrix is called normal if . Let be a normal complex matrix.
(i) Show that for any vector ,
(ii) Show that is also normal for any , where denotes the identity matrix.
(iii) Show that if is an eigenvector of with respect to the eigenvalue , then is also an eigenvector of , and determine the corresponding eigenvalue.
(iv) Show that if and are eigenvectors of with respect to distinct eigenvalues and respectively, then and are orthogonal.
(v) Show that if has a basis of eigenvectors, then can be diagonalised using an orthonormal basis. Justify your answer.
[You may use standard results provided that they are clearly stated.]
(b) Show that any matrix satisfying is normal, and deduce using results from (a) that its eigenvalues are real.
(c) Show that any matrix satisfying is normal, and deduce using results from (a) that its eigenvalues are purely imaginary.
(d) Show that any matrix satisfying is normal, and deduce using results from (a) that its eigenvalues have unit modulus.
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Paper 3, Section I, G
2015 commentDefine what is meant by a uniformly continuous function on a subset of a metric space. Show that every continuous function on a closed, bounded interval is uniformly continuous. [You may assume the Bolzano-Weierstrass theorem.]
Suppose that a function is continuous and tends to a finite limit at . Is necessarily uniformly continuous on Give a proof or a counterexample as appropriate.
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Paper 4, Section I, G
2015 commentDefine what is meant for two norms on a vector space to be Lipschitz equivalent.
Let denote the vector space of continuous functions with continuous first derivatives and such that for in some neighbourhood of the end-points and 1 . Which of the following four functions define norms on (give a brief explanation)?
Among those that define norms, which pairs are Lipschitz equivalent? Justify your answer.
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Paper 2, Section I, G
2015 commentShow that the map given by
is differentiable everywhere and find its derivative.
Stating accurately any theorem that you require, show that has a differentiable local inverse at a point if and only if
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Paper 1, Section II, G
2015 commentDefine what it means for a sequence of functions to converge uniformly on to a function .
Let , where are positive constants. Determine all the values of for which converges pointwise on . Determine all the values of for which converges uniformly on .
Let now . Determine whether or not converges uniformly on .
Let be a continuous function. Show that the sequence is uniformly convergent on if and only if .
[If you use any theorems about uniform convergence, you should prove these.]
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Paper 4, Section II, G
2015 commentConsider the space of bounded real sequences with the norm . Show that for every bounded sequence in there is a subsequence which converges in every coordinate, i.e. the sequence of real numbers converges for each . Does every bounded sequence in have a convergent subsequence? Justify your answer.
Let be the subspace of real sequences such that converges. Is complete in the norm (restricted from to ? Justify your answer.
Suppose that is a real sequence such that, for every , the series converges. Show that
Suppose now that is a real sequence such that, for every , the series converges. Show that
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Paper 3, Section II, G
2015 commentDefine what it means for a function to be differentiable at with derivative .
State and prove the chain rule for the derivative of , where is a differentiable function.
Now let be a differentiable function and let where is a constant. Show that is differentiable and find its derivative in terms of the partial derivatives of . Show that if holds everywhere in , then for some differentiable function
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Paper 2, Section II, G
2015 commentLet be normed spaces with norms . Show that for a map and , the following two statements are equivalent:
(i) For every given there exists such that whenever
(ii) for each sequence .
We say that is continuous at if (i), or equivalently (ii), holds.
Let now be a normed space. Let be a non-empty closed subset and define . Show that
In the case when with the standard Euclidean norm, show that there exists such that .
Let be two disjoint closed sets in . Must there exist disjoint open sets such that and ? Must there exist and such that for all and ? For each answer, give a proof or counterexample as appropriate.
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Paper 4, Section I, G
2015 commentLet be a continuous function defined on a connected open set . Prove carefully that the following statements are equivalent.
(i) There exists a holomorphic function on such that .
(ii) holds for every closed curve in .
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Paper 2, Section II, B
2015 comment(i) A function has a pole of order at . Derive a general expression for the residue of at involving and its derivatives.
(ii) Using contour integration along a contour in the upper half-plane, determine the value of the integral
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Paper 1, Section II, B
2015 comment(i) Show that transformations of the complex plane of the form
always map circles and lines to circles and lines, where and are complex numbers such that .
(ii) Show that the transformation
maps the unit disk centered at onto itself.
(iii) Deduce a conformal transformation that maps the non-concentric annular domain , to a concentric annular domain.
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Paper 3, Section II, G
2015 commentState the argument principle.
Let be an open set and a holomorphic injective function. Show that for each in and that is open.
Stating clearly any theorems that you require, show that for each and a sufficiently small ,
defines a holomorphic function on some open disc about .
Show that is the inverse for the restriction of to .
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Paper 1, Section I, B
2015 commentConsider the analytic (holomorphic) functions and on a nonempty domain where is nowhere zero. Prove that if for all in then there exists a real constant such that for all in .
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Paper 3, Section I, B
2015 commentFind the Fourier transform of the function
using an appropriate contour integration. Hence find the Fourier transform of its derivative, , and evaluate the integral
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Paper 4, Section II, B
2015 comment(i) State and prove the convolution theorem for Laplace transforms of two realvalued functions.
(ii) Let the function , be equal to 1 for and zero otherwise, where is a positive parameter. Calculate the Laplace transform of . Hence deduce the Laplace transform of the convolution . Invert this Laplace transform to obtain an explicit expression for .
[Hint: You may use the notation
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Paper 2, Section I, A
2015 commentIn a constant electric field a particle of rest mass and charge has position and velocity . At time , the particle is at rest at the origin. Including relativistic effects, calculate .
Sketch a graph of versus , commenting on the limit.
Calculate as an explicit function of and find the non-relativistic limit at small times .
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Paper 4, Section I, A
2015 commentFrom Maxwell's equations, derive the Biot-Savart law
giving the magnetic field produced by a steady current density that vanishes outside a bounded region .
[You may assume that you can choose a gauge such that the divergence of the magnetic vector potential is zero.]
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Paper 1, Section II, A
2015 comment(i) Write down the Lorentz force law for due to an electric field and magnetic field acting on a particle of charge moving with velocity .
(ii) Write down Maxwell's equations in terms of (the speed of light in a vacuum), in the absence of charges and currents.
(iii) Show that they can be manipulated into a wave equation for each component of .
(iv) Show that Maxwell's equations admit solutions of the form
where and are constant vectors and is a constant (all real). Derive a condition on and relate and .
(v) Suppose that a perfect conductor occupies the region and that a plane wave with is incident from the vacuum region . Write down boundary conditions for the and fields. Show that they can be satisfied if a suitable reflected wave is present, and determine the total and fields in real form.
(vi) At time , a particle of charge and mass is at moving with velocity . You may assume that the particle is far enough away from the conductor so that we can ignore its effect upon the conductor and that . Give a unit vector for the direction of the Lorentz force on the particle at time .
(vii) Ignoring relativistic effects, find the magnitude of the particle's rate of change of velocity in terms of and at time . Why is this answer inaccurate?
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Paper 3, Section II, A
2015 commentA charge density fills the region of 3-dimensional space , where is the radial distance from the origin and is a constant. Compute the electric field in all regions of space in terms of , the total charge of the region. Sketch a graph of the magnitude of the electric field versus (assuming that ).
Now let . Derive the surface charge density in terms of and and explain how a finite surface charge density may be obtained in this limit. Sketch the magnitude of the electric field versus in this limit. Comment on any discontinuities, checking a standard result involving for this particular case.
A second shell of equal and opposite total charge is centred on the origin and has a radius . Sketch the electric potential of this system, assuming that it tends to 0 as .
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Paper 2, Section II, A
2015 commentConsider the magnetic field
where and are unit vectors in the and directions, respectively. Imposing that this satisfies the expected equations for a static magnetic field in a vacuum, find and .
A circular wire loop of radius , mass and resistance lies in the plane with its centre on the -axis at and a magnetic field as given above. Calculate the magnetic flux through the loop arising from this magnetic field and also the force acting on the loop when a current is flowing around the loop in a clockwise direction about the -axis.
At , the centre of the loop is at the origin, travelling with velocity , where . Ignoring gravity and relativistic effects, and assuming that is only the induced current, find the time taken for the speed to halve in terms of and . By what factor does the rate of heat generation change in this time?
Where is the loop as as a function of
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Paper 1, Section I, B
2015 commentConsider a spherical bubble of radius in an inviscid fluid in the absence of gravity. The flow at infinity is at rest and the bubble undergoes translation with velocity . We assume that the flow is irrotational and derives from a potential given in spherical coordinates by
where is measured with respect to . Compute the force, , acting on the bubble. Show that the formula for can be interpreted as the acceleration force of a fraction of the fluid displaced by the bubble, and determine the value of .
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Paper 2, Section I, B
2015 commentConsider the two-dimensional velocity field with
(i) Show that the flow is incompressible and irrotational.
(ii) Derive the velocity potential, , and the streamfunction, .
(iii) Plot all streamlines passing through the origin.
(iv) Show that the complex function (where ) can be written solely as a function of the complex coordinate and determine that function.
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Paper 1, Section II, B
2015 commentA fluid layer of depth and dynamic viscosity is located underneath a fluid layer of depth and dynamic viscosity . The total fluid system of depth is positioned between a stationary rigid plate at and a rigid plate at moving with speed , where is constant. Ignore the effects of gravity.
(i) Using dimensional analysis only, and the fact that the stress should be linear in , derive the expected form of the shear stress acted by the fluid on the plate at as a function of and .
(ii) Solve for the unidirectional velocity profile between the two plates. State clearly all boundary conditions you are using to solve this problem.
(iii) Compute the exact value of the shear stress acted by the fluid on the plate at . Compare with the results in (i).
(iv) What is the condition on the viscosity of the bottom layer, , for the stress in (iii) to be smaller than it would be if the fluid had constant viscosity in both layers?
(v) Show that the stress acting on the plate at is equal and opposite to the stress on the plate at and justify this result physically.
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Paper 4, Section II, B
2015 commentConsider a steady inviscid, incompressible fluid of constant density in the absence of external body forces. A cylindrical jet of area and speed impinges fully on a stationary sphere of radius with . The flow is assumed to remain axisymmetric and be deflected into a conical sheet of vertex angle .
(i) Show that the speed of the fluid in the conical sheet is constant.
(ii) Use conservation of mass to show that the width of the fluid sheet at a distance from point of impact is given by
(iii) Use Euler's equation to derive the momentum integral equation
for a closed surface with normal where is the th component of the velocity field in cartesian coordinates and is the pressure.
(iv) Use the result from (iii) to calculate the net force on the sphere.
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Paper 3, Section II, B
2015 commentA source of sound induces a travelling wave of pressure above the free surface of a fluid located in the domain as
with . Here and are fixed real numbers. We assume that the flow induced in the fluid is irrotational.
(i) State the linearized equation of motion for the fluid and the free surface, , with all boundary conditions.
(ii) Solve for the velocity potential, , and the height of the free surface, . Verify that your solutions are dimensionally correct.
(iii) Interpret physically the behaviour of the solution when .
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Paper 1, Section I, F
2015 comment(i) Give a model for the hyperbolic plane. In this choice of model, describe hyperbolic lines.
Show that if are two hyperbolic lines and are points, then there exists an isometry of the hyperbolic plane such that and .
(ii) Let be a triangle in the hyperbolic plane with angles and . What is the area of ?
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Paper 3, Section , F
2015 commentState the sine rule for spherical triangles.
Let be a spherical triangle with vertices , and , with angles and at the respective vertices. Let , and be the lengths of the edges and respectively. Show that if and only if . [You may use the cosine rule for spherical triangles.] Show that this holds if and only if there exists a reflection such that and .
Are there equilateral triangles on the sphere? Justify your answer.
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Paper 3, Section II, F
2015 commentLet be a Möbius transformation on the Riemann sphere .
(i) Show that has either one or two fixed points.
(ii) Show that if is a Möbius transformation corresponding to (under stereographic projection) a rotation of through some fixed non-zero angle, then has two fixed points, , with .
(iii) Suppose has two fixed points with . Show that either corresponds to a rotation as in (ii), or one of the fixed points, say , is attractive, i.e. as for any .
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Paper 2, Section II, F
2015 comment(a) For each of the following subsets of , explain briefly why it is a smooth embedded surface or why it is not.
(b) Let be given by
and let . You may assume that is a smooth embedded surface.
Find the first fundamental form of this surface.
Find the second fundamental form of this surface.
Compute the Gaussian curvature of this surface.
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Paper 4, Section II, F
2015 commentLet be a curve in parameterized by arc length, and consider the surface of revolution in defined by the parameterization
In what follows, you may use that a curve in , with , is a geodesic if and only if
(i) Write down the first fundamental form for , and use this to write down a formula which is equivalent to being a unit speed curve.
(ii) Show that for a given , the circle on determined by is a geodesic if and only if .
(iii) Let be a curve in such that parameterizes a unit speed curve that is a geodesic in . For a given time , let denote the angle between the curve and the circle on determined by . Derive Clairault's relation that
is independent of .
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Paper 3, Section I, F
2015 commentState two equivalent conditions for a commutative ring to be Noetherian, and prove they are equivalent. Give an example of a ring which is not Noetherian, and explain why it is not Noetherian.
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Paper 4, Section I,
2015 commentLet be a commutative ring. Define what it means for an ideal to be prime. Show that is prime if and only if is an integral domain.
Give an example of an integral domain and an ideal , such that is not an integral domain.
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Paper 2, Section ,
2015 commentGive four non-isomorphic groups of order 12 , and explain why they are not isomorphic.
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Paper 1, Section II, F
2015 comment(i) Give the definition of a -Sylow subgroup of a group.
(ii) Let be a group of order . Show that there are at most two possibilities for the number of 3-Sylow subgroups, and give the possible numbers of 3-Sylow subgroups.
(iii) Continuing with a group of order 2835 , show that is not simple.
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Paper 4, Section II, F
2015 commentFind such that is a field . Show that for your choice of , every element of has a cube root in the field .
Show that if is a finite field, then the multiplicative group is cyclic.
Show that is a field. How many elements does have? Find a generator for .
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Paper 3, Section II, F
2015 commentCan a group of order 55 have 20 elements of order 11? If so, give an example. If not, give a proof, including the proof of any statements you need.
Let be a group of order , with and primes, . Suppose furthermore that does not divide . Show that is cyclic.
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Paper 2, Section II, F
2015 comment(a) Consider the homomorphism given by
Describe the image of this homomorphism as an abstract abelian group. Describe the quotient of by the image of this homomorphism as an abstract abelian group.
(b) Give the definition of a Euclidean domain.
Fix a prime and consider the subring of the rational numbers defined by
where 'gcd' stands for the greatest common divisor. Show that is a Euclidean domain.
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Paper 4, Section I, E
2015 commentDefine the dual space of a vector space . Given a basis of define its dual and show it is a basis of .
Let be a 3-dimensional vector space over and let be the basis of dual to the basis for . Determine, in terms of the , the bases dual to each of the following: (a) , (b) .
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Paper 2, Section I,
2015 commentLet denote a quadratic form on a real vector space . Define the rank and signature of .
Find the rank and signature of the following quadratic forms. (a) . (b) .
(c) .
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Paper 1, Section I, E
2015 commentLet and be finite dimensional vector spaces and a linear map. Suppose is a subspace of . Prove that
where denotes the rank of and denotes the restriction of to . Give examples showing that each inequality can be both a strict inequality and an equality.
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Paper 1, Section II, E
2015 commentDetermine the characteristic polynomial of the matrix
For which values of is invertible? When is not invertible determine (i) the Jordan normal form of , (ii) the minimal polynomial of .
Find a basis of such that is the matrix representing the endomorphism in this basis. Give a change of basis matrix such that .
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Paper 4, Section II, E
2015 commentSuppose and are subspaces of a vector space . Explain what is meant by and and show that both of these are subspaces of .
Show that if and are subspaces of a finite dimensional space then
Determine the dimension of the subspace of spanned by the vectors
Write down a matrix which defines a linear map with in the kernel and with image .
What is the dimension of the space spanned by all linear maps
(i) with in the kernel and with image contained in ,
(ii) with in the kernel or with image contained in ?
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Paper 3, Section II, E
2015 commentLet be matrices over a field . We say are simultaneously diagonalisable if there exists an invertible matrix such that is diagonal for all . We say the matrices are commuting if for all .
(i) Suppose are simultaneously diagonalisable. Prove that they are commuting.
(ii) Define an eigenspace of a matrix. Suppose are commuting matrices over a field . Let denote an eigenspace of . Prove that for all .
(iii) Suppose are commuting diagonalisable matrices. Prove that they are simultaneously diagonalisable.
(iv) Are the diagonalisable matrices over simultaneously diagonalisable? Explain your answer.
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Paper 2, Section II, E
2015 comment(i) Suppose is a matrix that does not have as an eigenvalue. Show that is non-singular. Further, show that commutes with .
(ii) A matrix is called skew-symmetric if . Show that a real skewsymmetric matrix does not have as an eigenvalue.
(iii) Suppose is a real skew-symmetric matrix. Show that is orthogonal with determinant 1 .
(iv) Verify that every orthogonal matrix with determinant 1 which does not have as an eigenvalue can be expressed as where is a real skew-symmetric matrix.
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Paper 4, Section I, H
2015 commentLet be independent identically distributed random variables with . Let , where is a constant. For each of the following cases, determine whether or not is a Markov chain: (a) ; (b) ; (c) .
In each case, if is a Markov chain, explain why, and give its state space and transition matrix; if it is not a Markov chain, give an example to demonstrate that it is not.
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Paper 3, Section I, H
2015 commentDefine what is meant by a communicating class and a closed class in a Markov chain.
A Markov chain with state space has transition matrix
Write down the communicating classes for this Markov chain and state whether or not each class is closed.
If , let be the smallest such that . Find for and . Describe the evolution of the chain if .
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Paper 2, Section II, H
2015 comment(a) What does it mean for a transition matrix and a distribution to be in detailed balance? Show that if and are in detailed balance then .
(b) A mathematician owns bicycles, which she sometimes uses for her journey from the station to College in the morning and for the return journey in the evening. If it is fine weather when she starts a journey, and if there is a bicycle available at the current location, then she cycles; otherwise she takes the bus. Assume that with probability , , it is fine when she starts a journey, independently of all other journeys. Let denote the number of bicycles at the current location, just before the mathematician starts the th journey.
(i) Show that is a Markov chain and write down its transition matrix.
(ii) Find the invariant distribution of the Markov chain.
(iii) Show that the Markov chain satisfies the necessary conditions for the convergence theorem for Markov chains and find the limiting probability that the mathematician's th journey is by bicycle.
[Results from the course may be used without proof provided that they are clearly stated.]
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Paper 1, Section II, H
2015 commentConsider a particle moving between the vertices of the graph below, taking steps along the edges. Let be the position of the particle at time . At time the particle moves to one of the vertices adjoining , with each of the adjoining vertices being equally likely, independently of previous moves. Explain briefly why is a Markov chain on the vertices. Is this chain irreducible? Find an invariant distribution for this chain.

Suppose that the particle starts at . By adapting the transition matrix, or otherwise, find the probability that the particle hits vertex before vertex .
Find the expected first passage time from to given no intermediate visit to .
[Results from the course may be used without proof provided that they are clearly stated.]
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Paper 4, Section I, 5C
2015 comment(a) The convolution of two functions is related to their Fourier transforms by
Derive Parseval's theorem for Fourier transforms from this relation.
(b) Let and
(i) Calculate the Fourier transform of .
(ii) Determine how the behaviour of in the limit depends on the value of . Briefly interpret the result.
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Paper 2, Section I, C
2015 comment(i) Write down the trigonometric form for the Fourier series and its coefficients for a function extended to a -periodic function on .
(ii) Calculate the Fourier series on of the function where is a real constant. Take the limit with in the coefficients of this series and briefly interpret the resulting expression.
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Paper 3, Section I,
2015 comment(a) From the defining property of the function,
for any function , prove that
(i)
(ii) for ,
(iii) If is smooth and has isolated zeros where the derivative , then
(b) Show that the function defined by
is the function.
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Paper 1, Section II, C
2015 comment(i) Briefly describe the Sturm-Liouville form of an eigenfunction equation for real valued functions with a linear, second-order ordinary differential operator. Briefly summarize the properties of the solutions.
(ii) Derive the condition for self-adjointness of the differential operator in (i) in terms of the boundary conditions of solutions to the Sturm-Liouville equation. Give at least three types of boundary conditions for which the condition for self-adjointness is satisfied.
(iii) Consider the inhomogeneous Sturm-Liouville equation with weighted linear term
on the interval , where and are real functions on and is the weighting function. Let be a Green's function satisfying
Let solutions and the Green's function satisfy the same boundary conditions of the form at at are not both zero and are not both zero) and likewise for for the same constants and . Show that the Sturm-Liouville equation can be written as a so-called Fredholm integral equation of the form
where and depends on and the forcing term . Write down in terms of an integral involving and .
(iv) Derive the Fredholm integral equation for the Sturm-Liouville equation on the interval
with .
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Paper 3, Section II, C
2015 comment(i) Consider the Poisson equation with forcing term on the infinite domain with . Derive the Green's function for this equation using the divergence theorem. [You may assume without proof that the divergence theorem is valid for the Green's function.]
(ii) Consider the Helmholtz equation
where is a real constant. A Green's function for this equation can be constructed from of (i) by assuming where and is a regular function. Show that and that satisfies the equation
(iii) Take the Green's function with the specific solution to Eq. ( ) and consider the Helmholtz equation on the semi-infinite domain . Use the method of images to construct a Green's function for this problem that satisfies the boundary conditions
(iv) A solution to the Helmholtz equation on a bounded domain can be constructed in complete analogy to that of the Poisson equation using the Green's function in Green's 3rd identity
where denotes the volume of the domain, its boundary and the outgoing normal derivative on the boundary. Now consider the homogeneous Helmholtz equation on the domain with boundary conditions at and
where and and are real constants. Construct a solution in integral form to this equation using cylindrical coordinates with .
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Paper 2, Section II, C
2015 comment(i) The Laplace operator in spherical coordinates is
Show that general, regular axisymmetric solutions to the equation are given by
where are constants and are the Legendre polynomials. [You may use without proof that regular solutions to Legendre's equation are given by with and non-negative integer .]
(ii) Consider a uniformly charged wire in the form of a ring of infinitesimal width with radius and a constant charge per unit length . By Coulomb's law, the electric potential due to a point charge at a point a distance from the charge is
where is a constant. Let the -axis be perpendicular to the circle and pass through the circle's centre (see figure). Show that the potential due to the charged ring at a point on the -axis at location is given by

(iii) The potential generated by the charged ring of (ii) at arbitrary points (excluding points directly on the ring which can be ignored for this question) is determined by Laplace's equation . Calculate this potential with the boundary condition , where . [You may use without proof that
for . Furthermore, the Legendre polynomials are normalized such that
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Paper 4, Section II, 17C
2015 commentDescribe the method of characteristics to construct solutions for 1st-order, homogeneous, linear partial differential equations
with initial data prescribed on a curve .
Consider the partial differential equation (here the two independent variables are time and spatial direction )
with initial data .
(i) Calculate the characteristic curves of this equation and show that remains constant along these curves. Qualitatively sketch the characteristics in the diagram, i.e. the axis is the horizontal and the axis is the vertical axis.
(ii) Let denote the value of a characteristic at time and thus label the characteristic curves. Let denote the value at time of a characteristic with given . Show that becomes a non-monotonic function of (at fixed ) at times , i.e. has a local minimum or maximum. Qualitatively sketch snapshots of the solution for a few fixed values of and briefly interpret the onset of the non-monotonic behaviour of at .
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Paper 3, Section I,
2015 commentDefine what it means for a topological space to be (i) connected (ii) path-connected.
Prove that any path-connected space is connected. [You may assume the interval is connected.
Give a counterexample (without justification) to the converse statement.
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Paper 2, Section I,
2015 commentLet and be topological spaces and a continuous map. Suppose is a subset of such that is closed (where denotes the closure of ). Prove that
Give an example where and are as above but is not closed.
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Paper 1, Section II, E
2015 commentGive the definition of a metric on a set and explain how this defines a topology on .
Suppose is a metric space and is an open set in . Let and such that the open ball and . Prove that .
Explain what it means (i) for a set to be dense in , (ii) to say is a base for a topology .
Prove that any metric space which contains a countable dense set has a countable basis.
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Paper 4, Section II, E
2015 commentExplain what it means for a metric space to be (i) compact, (ii) sequentially compact. Prove that a compact metric space is sequentially compact, stating clearly any results that you use.
Let be a compact metric space and suppose satisfies for all . Prove that is surjective, stating clearly any results that you use. [Hint: Consider the sequence for .]
Give an example to show that the result does not hold if is not compact.
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Paper 1, Section I, 6D
2015 commentLet
where is a real parameter. Find the factorization of the matrix . Give the constraint on for A to be positive definite.
For , use this factorization to solve the system via forward and backward substitution.
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Paper 4, Section I, D
2015 commentGiven distinct points , let be the real polynomial of degree that interpolates a continuous function at these points. State the Lagrange interpolation formula.
Prove that can be written in the Newton form
where is the divided difference, which you should define. [An explicit expression for the divided difference is not required.]
Explain why it can be more efficient to use the Newton form rather than the Lagrange formula.
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Paper 1, Section II, 18D
2015 commentDetermine the real coefficients such that
is exact when is any real polynomial of degree 2 . Check explicitly that the quadrature is exact for with these coefficients.
State the Peano kernel theorem and define the Peano kernel . Use this theorem to show that if , and are chosen as above, then
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Paper 3, Section II, D
2015 commentDefine the QR factorization of an matrix . Explain how it can be used to solve the least squares problem of finding the vector which minimises , where , and is the Euclidean norm.
Explain how to construct and by the Gram-Schmidt procedure. Why is this procedure not useful for numerical factorization of large matrices?
Let
Using the Gram-Schmidt procedure find a QR decomposition of A. Hence solve the least squares problem giving both and .
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Paper 2, Section II, D
2015 commentDefine the linear stability domain for a numerical method to solve . What is meant by an -stable method? Briefly explain the relevance of these concepts in the numerical solution of ordinary differential equations.
Consider
where . What is the order of this method?
Find the linear stability domain of this method. For what values of is the method A-stable?
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Paper 1, Section I, H
2015 comment(a) Consider a network with vertices in and directed edges in . Suppose that 1 is the source and is the sink. Let , be the capacity of the edge from vertex to vertex for . Let a cut be a partition of into and with and . Define the capacity of the cut . Write down the maximum flow problem. Prove that the maximum flow is bounded above by the minimum cut capacity.
(b) Find the maximum flow from the source to the sink in the network below, where the directions and capacities of the edges are shown. Explain your reasoning.

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Paper 2, Section I, H
2015 commentDefine what it means to say that a set is convex. What is meant by an extreme point of a convex set ?
Consider the set given by
Show that is convex, and give the coordinates of all extreme points of .
For all possible choices of and , find the maximum value of subject to .
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Paper 4, Section II, 20H
2015 commentSuppose the recycling manager in a particular region is responsible for allocating all the recyclable waste that is collected in towns in the region to the recycling centres in the region. Town produces lorry loads of recyclable waste each day, and recycling centre needs to handle lorry loads of waste a day in order to be viable. Suppose that . Suppose further that is the cost of transporting a lorry load of waste from town to recycling centre . The manager wishes to decide the number of lorry loads of recyclable waste that should go from town to recycling centre , , in such a way that all the recyclable waste produced by each town is transported to recycling centres each day, and each recycling centre works exactly at the viable level each day. Use the Lagrangian sufficiency theorem, which you should quote carefully, to derive necessary and sufficient conditions for to minimise the total cost under the above constraints.
Suppose that there are three recycling centres and , needing 5,20 and 20 lorry loads of waste each day, respectively, and suppose there are three towns and producing 20,15 and 10 lorry loads of waste each day, respectively. The costs of transporting a lorry load of waste from town to recycling centres and are and , respectively. The corresponding costs for town are and , while for town they are and . Recycling centre has reported that it currently receives 5 lorry loads of waste per day from town , and recycling centre has reported that it currently receives 10 lorry loads of waste per day from each of towns and c. Recycling centre has failed to report. What is the cost of the current arrangement for transporting waste from the towns to the recycling centres? Starting with the current arrangement as an initial solution, use the transportation algorithm (explaining each step carefully) in order to advise the recycling manager how many lorry loads of waste should go from each town to each of the recycling centres in order to minimise the cost. What is the minimum cost?
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Paper 3, Section II, H
2015 commentConsider the linear programming problem :
where and are in is a real matrix, is in and denotes transpose. Derive the dual linear programming problem . Show from first principles that the dual of is .
Suppose that and . Write down the dual and find the optimal solution of the dual using the simplex algorithm. Hence, or otherwise, find the optimal solution of .
Suppose that is changed to . Give necessary and sufficient conditions for still to be the optimal solution of . If , find the range of values for for which is still the optimal solution of .
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Paper 4, Section I, D
2015 commentThe radial wavefunction for an electron in a hydrogen atom satisfies the equation
Briefly explain the origin of each term in this equation.
The wavefunctions for the ground state and the first radially excited state, both with , can be written as
where and are normalisation constants. Verify that is a solution of , determining and finding the corresponding energy eigenvalue . Assuming that is a solution of , compare coefficients of the dominant terms when is large to determine the corresponding energy eigenvalue . [You do not need to find or , nor show that is a solution of
A hydrogen atom makes a transition from the first radially excited state to the ground state, emitting a photon. What is the angular frequency of the emitted photon?
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Paper 3, Section , D
2015 commentA quantum-mechanical system has normalised energy eigenstates and with non-degenerate energies and respectively. The observable has normalised eigenstates,
where is a positive real constant. Determine .
Initially, at time , the state of the system is . Write down an expression for , the state of the system with . What is the probability that a measurement of energy at time will yield ?
For the same initial state, determine the probability that a measurement of at time will yield and the probability that it will yield .
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Paper 1, Section II, D
2015 commentWrite down expressions for the probability density and the probability current for a particle in one dimension with wavefunction . If obeys the timedependent Schrödinger equation with a real potential, show that
Consider a stationary state, , with
where are real. Evaluate for this state in the regimes and .
Consider a real potential,
where is the Dirac delta function, and . Assuming that is continuous at , derive an expression for
Hence calculate the reflection and transmission probabilities for a particle incident from with energy
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Paper 3, Section II, D
2015 commentDefine the angular momentum operators for a particle in three dimensions in terms of the position and momentum operators and . Write down an expression for and use this to show that where . What is the significance of these two commutation relations?
Let be both an eigenstate of with eigenvalue zero and an eigenstate of with eigenvalue . Show that is also an eigenstate of both and and determine the corresponding eigenvalues.
Find real constants and such that
is an eigenfunction of with eigenvalue zero and an eigenfunction of with an eigenvalue which you should determine. [Hint: You might like to show that
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Paper 2, Section II, D
2015 commentA quantum-mechanical harmonic oscillator has Hamiltonian
where is a positive real constant. Show that and are Hermitian operators.
The eigenfunctions of can be written as
where is a polynomial of degree with even (odd) parity for even (odd) and . Show that for all of the states .
State the Heisenberg uncertainty principle and verify it for the state by computing and . [Hint: You should properly normalise the state.]
The oscillator is in its ground state when the potential is suddenly changed so that . If the wavefunction is expanded in terms of the energy eigenfunctions of the new Hamiltonian, , what can be said about the coefficient of for odd ? What is the probability that the particle is in the new ground state just after the change?
[Hint: You may assume that if then and .]
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Paper 1, Section I, H
2015 commentSuppose that are independent normally distributed random variables, each with mean and variance 1 , and consider testing against . Explain what is meant by the critical region, the size and the power of a test.
For , derive the test that is most powerful among all tests of size at most . Obtain an expression for the power of your test in terms of the standard normal distribution function .
[Results from the course may be used without proof provided they are clearly stated.]
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Paper 2, Section I, H
2015 commentSuppose that, given , the random variable has , Suppose that the prior density of is , for some known . Derive the posterior density of based on the observation .
For a given loss function , a statistician wants to calculate the value of that minimises the expected posterior loss
Suppose that . Find in terms of in the following cases:
(a) ;
(b) .
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Paper 4, Section II, H
2015 commentConsider a linear model where is an vector of observations, is a known matrix, is a vector of unknown parameters and is an vector of independent normally distributed random variables each with mean zero and unknown variance . Write down the log-likelihood and show that the maximum likelihood estimators and of and respectively satisfy
denotes the transpose . Assuming that is invertible, find the solutions and of these equations and write down their distributions.
Prove that and are independent.
Consider the model and . Suppose that, for all and , and that , are independent random variables where is unknown. Show how this model may be written as a linear model and write down and . Find the maximum likelihood estimators of and in terms of the . Derive a confidence interval for and for .
[You may assume that, if is multivariate normal with , then and are independent.]
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Paper 1, Section II, H
2015 commentSuppose are independent identically distributed random variables each with probability mass function , where is an unknown parameter. State what is meant by a sufficient statistic for . State the factorisation criterion for a sufficient statistic. State and prove the Rao-Blackwell theorem.
Suppose that are independent identically distributed random variables with
where is a known positive integer and is unknown. Show that is unbiased for .
Show that is sufficient for and use the Rao-Blackwell theorem to find another unbiased estimator for , giving details of your derivation. Calculate the variance of and compare it to the variance of .
A statistician cannot remember the exact statement of the Rao-Blackwell theorem and calculates in an attempt to find an estimator of . Comment on the suitability or otherwise of this approach, giving your reasons.
[Hint: If and are positive integers then, for
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Paper 3, Section II, H
2015 comment(a) Suppose that are independent identically distributed random variables, each with density for some unknown . Use the generalised likelihood ratio to obtain a size test of against .
(b) A die is loaded so that, if is the probability of face , then , and . The die is thrown times and face is observed times. Write down the likelihood function for and find the maximum likelihood estimate of .
Consider testing whether or not for this die. Find the generalised likelihood ratio statistic and show that
where you should specify and in terms of . Explain how to obtain an approximate size test using the value of . Explain what you would conclude (and why ) if .
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Paper 1, Section I, A
2015 commentConsider a frictionless bead on a stationary wire. The bead moves under the action of gravity acting in the negative -direction and the wire traces out a path , connecting points and . Using a first integral of the Euler-Lagrange equations, find the choice of which gives the shortest travel time, starting from rest. You may give your solution for and separately, in parametric form.
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Paper 3, Section , A
2015 comment(a) Define what it means for a function to be convex.
(b) Define the Legendre transform of a convex function , where . Show that is a convex function.
(c) Find the Legendre transform of the function , and the domain of .
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Paper 4, Section II, F
2015 comment(i) Explain how a linear system on a curve may induce a morphism from to projective space. What condition on the linear system is necessary to yield a morphism such that the pull-back of a hyperplane section is an element of the linear system? What condition is necessary to imply the morphism is an embedding?
(ii) State the Riemann-Roch theorem for curves.
(iii) Show that any divisor of degree 5 on a curve of genus 2 induces an embedding.
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Paper 3, Section II, F
2015 comment(i) Let be an affine variety. Define the tangent space of at a point . Say what it means for the variety to be singular at .
(ii) Find the singularities of the surface in given by the equation
(iii) Consider . Let be the blowup of the origin. Compute the proper transform of in , and show it is non-singular.
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Paper 2, Section II, F
2015 comment(i) Define the radical of an ideal.
(ii) Assume the following statement: If is an algebraically closed field and is an ideal, then either or . Prove the Hilbert Nullstellensatz, namely that if with algebraically closed, then
(iii) Show that if is a commutative ring and are ideals, then
(iv) Is
Give a proof or a counterexample.
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Paper 1, Section II, F
2015 commentLet be an algebraically closed field.
(i) Let and be affine varieties defined over . Given a map , define what it means for to be a morphism of affine varieties.
(ii) With still affine varieties over , show that there is a one-to-one correspondence between , the set of morphisms between and , and , the set of -algebra homomorphisms between and .
(iii) Let be given by . Show that the image of is an affine variety , and find a set of generators for .
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Paper 3, Section II, H
2015 commentLet and be simplicial complexes. Explain what is meant by a simplicial approximation to a continuous map . State the simplicial approximation theorem, and define the homomorphism induced on homology by a continuous map between triangulable spaces. [You do not need to show that the homomorphism is welldefined.]
Let be given by for a positive integer , where is considered as the unit complex numbers. Compute the map induced by on homology.
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Paper 4, Section II, H
2015 commentState the Mayer-Vietoris theorem for a simplicial complex which is the union of two subcomplexes and . Explain briefly how the connecting homomorphism is defined.
If is the union of subcomplexes , with , such that each intersection
is either empty or has the homology of a point, then show that
Construct examples for each showing that this is sharp.
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Paper 2, Section II, H
2015 commentDefine what it means for to be a covering map, and what it means to say that is a universal cover.
Let be a universal cover, be a locally path connected subspace, and be a path component containing a point with . Show that the restriction is a covering map, and that under the Galois correspondence it corresponds to the subgroup
of .
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Paper 1, Section II, H
2015 commentState carefully a version of the Seifert-van Kampen theorem for a cover of a space by two closed sets.
Let be the space obtained by gluing together a Möbius band and a torus along a homeomorphism of the boundary of with . Find a presentation for the fundamental group of , and hence show that it is infinite and non-abelian.
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Paper 4, Section II,
2015 commentLet be a Bravais lattice with basis vectors . Define the reciprocal lattice and write down basis vectors for in terms of the basis for .
A finite crystal consists of identical atoms at sites of given by
A particle of mass scatters off the crystal; its wavevector is before scattering and after scattering, with . Show that the scattering amplitude in the Born approximation has the form
where is the potential due to a single atom at the origin and depends on the crystal structure. [You may assume that in the Born approximation the amplitude for scattering off a potential is where tilde denotes the Fourier transform.]
Derive an expression for that is valid when . Show also that when is a reciprocal lattice vector is equal to the total number of atoms in the crystal. Comment briefly on the significance of these results.
Now suppose that is a face-centred-cubic lattice:
where is a constant. Show that for a particle incident with , enhanced scattering is possible for at least two values of the scattering angle, and , related by
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Paper 1, Section II, A
2015 commentDefine the Rayleigh-Ritz quotient for a normalisable state of a quantum system with Hamiltonian . Given that the spectrum of is discrete and that there is a unique ground state of energy , show that and that equality holds if and only if is the ground state.
A simple harmonic oscillator (SHO) is a particle of mass moving in one dimension subject to the potential
Estimate the ground state energy of the SHO by using the ground state wavefunction for a particle in an infinite potential well of width , centred on the origin (the potential is for and for . Take as the variational parameter.
Perform a similar estimate for the energy of the first excited state of the SHO by using the first excited state of the infinite potential well as a trial wavefunction.
Is the estimate for necessarily an upper bound? Justify your answer.
You may use : and
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Paper 3, Section II, A
2015 commentA particle of mass and energy moves in one dimension subject to a periodic potential
Determine the corresponding Floquet matrix . [You may assume without proof that for the Schrödinger equation with potential the wavefunction is continuous at and satisfies
Explain briefly, with reference to Bloch's theorem, how restrictions on the energy of a Bloch state can be derived from . Deduce that for the potential above, is confined to a range whose boundary values are determined by
Sketch the left-hand and right-hand sides of each of these equations as functions of . Hence show that there is exactly one allowed band of negative energies with either (i) or (ii) and determine the values of for which each of these cases arise. [You should not attempt to evaluate the constants ]
Comment briefly on the limit with fixed.
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Paper 2, Section II, A
2015 commentA beam of particles of mass and energy is incident on a target at the origin described by a spherically symmetric potential . Assuming the potential decays rapidly as , write down the asymptotic form of the wavefunction, defining the scattering amplitude .
Consider a free particle with energy . State, without proof, the general axisymmetric solution of the Schrödinger equation for in terms of spherical Bessel and Neumann functions and , and Legendre polynomials . Hence define the partial wave phase shifts for scattering from a potential and derive the partial wave expansion for in terms of phase shifts.
Now suppose
with . Show that the S-wave phase shift obeys
where . Deduce that for an S-wave solution
[You may assume :
and as
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Paper 4, Section II, K
2015 comment(i) Let be a Markov chain on and . Let be the hitting time of and denote the total time spent at by the chain before hitting . Show that if , then
(ii) Define the Moran model and show that if is the number of individuals carrying allele at time and is the fixation time of allele , then
Show that conditionally on fixation of an allele being present initially in individuals,
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Paper 3, Section II, K
2015 comment(i) Let be a Poisson process of parameter . Let be obtained by taking each point of and, independently of the other points, keeping it with probability . Show that is another Poisson process and find its intensity. Show that for every fixed the random variables and are independent.
(ii) Suppose we have bins, and balls arrive according to a Poisson process of rate 1 . Upon arrival we choose a bin uniformly at random and place the ball in it. We let be the maximum number of balls in any bin at time . Show that
[You may use the fact that if is a Poisson random variable of mean 1 , then
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Paper 2, Section II,
2015 comment(i) Defne a Poisson process on with rate . Let and be two independent Poisson processes on of rates and respectively. Prove that is also a Poisson process and find its rate.
(ii) Let be a discrete time Markov chain with transition matrix on the finite state space . Find the generator of the continuous time Markov chain in terms of and . Show that if is an invariant distribution for , then it is also invariant for .
Suppose that has an absorbing state . If and are the absorption times for and respectively, write an equation that relates and , where .
[Hint: You may want to prove that if are i.i.d. non-negative random variables with and is an independent non-negative random variable, then
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Paper 1, Section II, K
2015 comment(a) Give the definition of a birth and death chain in terms of its generator. Show that a measure is invariant for a birth and death chain if and only if it solves the detailed balance equations.
(b) There are servers in a post office and a single queue. Customers arrive as a Poisson process of rate and the service times at each server are independent and exponentially distributed with parameter . Let denote the number of customers in the post office at time . Find conditions on and for to be positive recurrent, null recurrent and transient, justifying your answers.
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Paper 4, Section II, C
2015 commentConsider the ordinary differential equation
where
and are constants. Look for solutions in the asymptotic form
and determine in terms of , as well as in terms of .
Deduce that the Bessel equation
where is a complex constant, has two solutions of the form
and determine and in terms of
Can the above asymptotic expansions be valid for all , or are they valid only in certain domains of the complex -plane? Justify your answer briefly.
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Paper 3, Section II,
2015 commentShow that
where is an integral from 0 to along the line and is an integral from 1 to along a steepest-descent contour which you should determine.
By employing in the integrals and the changes of variables and , respectively, compute the first two terms of the large asymptotic expansion of the integral above.
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Paper 1, Section II, C
2015 comment(a) State the integral expression for the gamma function , for , and express the integral
in terms of . Explain why the constraints on are necessary.
(b) Show that
for some constants and . Determine the constants and , and express in terms of the gamma function.
State without proof the basic result needed for the rigorous justification of the above asymptotic formula.
[You may use the identity:
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Paper 4, Section I, D
2015 commentA triatomic molecule is modelled by three masses moving in a line while connected to each other by two identical springs of force constant as shown in the figure.

(a) Write down the Lagrangian and derive the equations describing the motion of the atoms.
(b) Find the normal modes and their frequencies. What motion does the lowest frequency represent?
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Paper 3, Section I,
2015 comment(a) Consider a particle of mass that undergoes periodic motion in a one-dimensional potential . Write down the Hamiltonian for the system. Explain what is meant by the angle-action variables of the system and write down the integral expression for the action variable .
(b) For and fixed total energy , describe the shape of the trajectories in phase-space. By using the expression for the area enclosed by the trajectory, or otherwise, find the action variable in terms of and . Hence describe how changes with if varies slowly with time. Justify your answer.
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Paper 2, Section I, D
2015 commentThe Lagrangian for a heavy symmetric top of mass , pinned at a point that is a distance from the centre of mass, is
(a) Find all conserved quantities. In particular, show that , the spin of the top, is constant.
(b) Show that obeys the equation of motion
where the explicit form of should be determined.
(c) Determine the condition for uniform precession with no nutation, that is and const. For what values of does such uniform precession occur?
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Paper 1, Section I, D
2015 comment(a) The action for a one-dimensional dynamical system with a generalized coordinate and Lagrangian is given by
State the principle of least action and derive the Euler-Lagrange equation.
(b) A planar spring-pendulum consists of a light rod of length and a bead of mass , which is able to slide along the rod without friction and is attached to the ends of the rod by two identical springs of force constant as shown in the figure. The rod is pivoted at one end and is free to swing in a vertical plane under the influence of gravity.

(i) Identify suitable generalized coordinates and write down the Lagrangian of the system.
(ii) Derive the equations of motion.
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Paper 4, Section II, C
2015 commentConsider a rigid body with angular velocity , angular momentum and position vector , in its body frame.
(a) Use the expression for the kinetic energy of the body,
to derive an expression for the tensor of inertia of the body, I. Write down the relationship between and .
(b) Euler's equations of torque-free motion of a rigid body are
Working in the frame of the principal axes of inertia, use Euler's equations to show that the energy and the squared angular momentum are conserved.
(c) Consider a cuboid with sides and , and with mass distributed uniformly.
(i) Use the expression for the tensor of inertia derived in (a) to calculate the principal moments of inertia of the body.
(ii) Assume and , and suppose that the initial conditions are such that
with the initial angular velocity perpendicular to the intermediate principal axis . Derive the first order differential equation for in terms of and and hence determine the long-term behaviour of .
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Paper 2, Section II, C
2015 comment(a) Consider a Lagrangian dynamical system with one degree of freedom. Write down the expression for the Hamiltonian of the system in terms of the generalized velocity , momentum , and the Lagrangian . By considering the differential of the Hamiltonian, or otherwise, derive Hamilton's equations.
Show that if is ignorable (cyclic) with respect to the Lagrangian, i.e. , then it is also ignorable with respect to the Hamiltonian.
(b) A particle of charge and mass moves in the presence of electric and magnetic fields such that the scalar and vector potentials are and , where are Cartesian coordinates and are constants. The Lagrangian of the particle is
Starting with the Lagrangian, derive an explicit expression for the Hamiltonian and use Hamilton's equations to determine the motion of the particle.
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Paper 4, Section I, G
2015 commentExplain how to construct binary Reed-Muller codes. State and prove a result giving the minimum distance for each such Reed-Muller code.
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Paper 3, Section I, G
2015 commentLet be a random variable that takes each value in the finite alphabet with probability . Show that, if each is an integer greater than 0 and , then there is a decodable binary code with each codeword having length .
Prove that, for any decodable code , we have
where is the entropy of the random variable . When is there equality in this inequality?
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Paper 2, Section I, G
2015 commentA random variable takes values in the alphabet with probabilities and . Calculate the entropy of .
Define what it means for a code for a general finite alphabet to be optimal. Find such a code for the distribution above and show that there are optimal codes for this distribution with differing lengths of codeword.
[You may use any results from the course without proof. Note that .]
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Paper 1, Section I,
2015 commentLet be a finite alphabet. Explain what is meant by saying that a binary code has minimum distance . If is such a binary code with minimum distance , show that is error-detecting and error-correcting.
Show that it is possible to construct a code that has minimum distance for any integer .
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Paper 1, Section II, G
2015 commentDefine the Hamming code. Show that it is a perfect, linear, 1-error correcting code.
I wish to send a message through a noisy channel to a friend. The message consists of a large number of letters from a 16 -letter alphabet . When my friend has decoded the message, she can tell whether there have been any errors. If there have, she asks me to send the message again and this is repeated until she has received the message without error. For each individual binary digit that is transmitted, there is independently a small probability of an error.
(a) Suppose that I encode my message by writing each letter as a 4-bit binary string. The whole message is then bits long. What is the probability that the entire message is transmitted without error? How many times should I expect to transmit the message until my friend receives it without error?
(b) As an alternative, I use the Hamming code to encode each letter of as a 7-bit binary string. What is the probability that my friend can decode a single 7-bit string correctly? Deduce that the probability that the entire message is correctly decoded is given approximately by
Which coding method is better?
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Paper 2, Section II, G
2015 commentBriefly describe the public key cipher.
Just before it went into liquidation, the Internet Bank decided that it wanted to communicate with each of its customers using an RSA cipher. So, it chose a large modulus , which is the product of two large prime numbers, and chose encrypting exponents and decrypting exponents for each customer . The bank published and and sent the decrypting exponent secretly to customer . Show explicitly that the cipher can be broken by each customer.
The bank sent out the same message to each customer. I am not a customer of the bank but have two friends who are and I notice that their published encrypting exponents are coprime. Explain how I can find the original message. Can I break the cipher?
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Paper 4, Section I, C
2015 commentCalculate the total effective number of relativistic spin states present in the early universe when the temperature is if there are three species of low-mass neutrinos and antineutrinos in addition to photons, electrons and positrons. If the weak interaction rate is and the expansion rate of the universe is , where is the total density of the universe, calculate the temperature at which the neutrons and protons cease to interact via weak interactions, and show that .
State the formula for the equilibrium ratio of neutrons to protons at , and briefly describe the sequence of events as the temperature falls from to the temperature at which the nucleosynthesis of helium and deuterium ends.
What is the effect of an increase or decrease of on the abundance of helium-4 resulting from nucleosynthesis? Why do changes in have a very small effect on the final abundance of deuterium?
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Paper 3, Section I, C
2015 commentWhat is the flatness problem? Show by reference to the Friedmann equation how a period of accelerated expansion of the scale factor in the early stages of the universe can solve the flatness problem if , where is the mass density and is the pressure.
In the very early universe, where we can neglect the spatial curvature and the cosmological constant, there is a homogeneous scalar field with a vacuum potential energy
and the Friedmann energy equation (in units where ) is
where is the Hubble parameter. The field obeys the evolution equation
During inflation, evolves slowly after starting from a large initial value at . State what is meant by the slow-roll approximation. Show that in this approximation,
where is the initial value of .
As decreases from its initial value , what is its approximate value when the slow-roll approximation fails?
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Paper 2, Section I, C
2015 commentThe mass density perturbation equation for non-relativistic matter with wave number in the late universe is
Suppose that a non-relativistic fluid with the equation of state dominates the universe when , and the curvature and the cosmological constant can be neglected. Show that the sound speed can be written in the form where is a constant.
Find power-law solutions to of the form and hence show that the general solution is
where
Interpret your solutions in the two regimes and where .
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Paper 1, Section I, C
2015 commentConsider three galaxies and with position vectors and in a homogeneous universe. Assuming they move with non-relativistic velocities and , show that spatial homogeneity implies that the velocity field satisfies
and hence that is linearly related to by
where the components of the matrix are independent of .
Suppose the matrix has the form
with constant. Describe the kinematics of the cosmological expansion.
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Paper 3, Section II, C
2015 commentMassive particles and antiparticles each with mass and respective number densities and are present at time in the radiation era of an expanding universe with zero curvature and no cosmological constant. Assuming they interact with crosssection at speed , explain, by identifying the physical significance of each of the terms, why the evolution of is described by
where the expansion scale factor of the universe is , and where the meaning of should be briefly explained. Show that
Assuming initial particle-antiparticle symmetry, show that
where is the equilibrium number density at temperature .
Let and . Show that
where and is the Hubble expansion rate when .
When , the number density can be assumed to be depleted only by annihilations. If is constant, show that as at late time, approaches a constant value given by
Why do you expect weakly interacting particles to survive in greater numbers than strongly interacting particles?
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Paper 1, Section II, C
2015 commentA closed universe contains black-body radiation, has a positive cosmological constant , and is governed by the equation
where is the scale factor and is a positive constant. Using the substitution and the boundary condition , deduce the boundary condition for and show that
and hence that
Express the constant in terms of and .
Sketch the graphs of for the cases and .
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Paper 4, Section II, G
2015 commentLet denote the set of unitary complex matrices. Show that is a smooth (real) manifold, and find its dimension. [You may use any general results from the course provided they are stated correctly.] For any matrix in and an complex matrix, determine when represents a tangent vector to at .
Consider the tangent spaces to equipped with the metric induced from the standard (Euclidean) inner product on the real vector space of complex matrices, given by , where denotes the real part and denotes the conjugate transpose of . Suppose that represents a tangent vector to at the identity matrix . Sketch an explicit construction of a geodesic curve on passing through and with tangent direction , giving a brief proof that the acceleration of the curve is always orthogonal to the tangent space to .
[Hint: You will find it easier to work directly with complex matrices, rather than the corresponding real matrices.]
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Paper 3 , Section II, G
2015 commentShow that the surface of revolution in is homeomorphic to a cylinder and has everywhere negative Gaussian curvature. Show moreover the existence of a closed geodesic on .
Let be an arbitrary embedded surface which is homeomorphic to a cylinder and has everywhere negative Gaussian curvature. By using a suitable version of the Gauss-Bonnet theorem, show that contains at most one closed geodesic. [If required, appropriate forms of the Jordan curve theorem in the plane may also be used without proof.
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Paper 2, Section II, G
2015 commentIf denotes a domain in , what is meant by saying that a smooth map is an immersion? Define what it means for such an immersion to be isothermal. Explain what it means to say that an immersed surface is minimal.
Let be an isothermal immersion. Show that it is minimal if and only if are harmonic functions of . [You may use the formula for the mean curvature given in terms of the first and second fundamental forms, namely
Produce an example of an immersed minimal surface which is not an open subset of a catenoid, helicoid, or a plane. Prove that your example does give an immersed minimal surface in .
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Paper 1, Section II,
2015 commentLet be a domain (connected open subset) with boundary a continuously differentiable simple closed curve. Denoting by the area of and the length of the curve , state and prove the isoperimetric inequality relating and with optimal constant, including the characterization for equality. [You may appeal to Wirtinger's inequality as long as you state it precisely.]
Does the result continue to hold if the boundary is allowed finitely many points at which it is not differentiable? Briefly justify your answer by giving either a counterexample or an indication of a proof.
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Paper 4, Section II, B
2015 commentLet be a continuous one-dimensional map of an interval . Explain what is meant by the statements (i) that has a horseshoe and (ii) that is chaotic (according to Glendinning's definition).
Assume that has a 3-cycle with and, without loss of generality, . Prove that has a horseshoe. [You may assume the intermediate value theorem.]
Represent the effect of on the intervals and by means of a directed graph, explaining carefully how the graph is constructed. Explain what feature of the graph implies the existence of a 3-cycle.
The map has a 5-cycle with and , and . For which , is an -cycle of guaranteed to exist? Is guaranteed to be chaotic? Is guaranteed to have a horseshoe? Justify your answers. [You may use a suitable directed graph as part of your arguments.]
How do your answers to the above change if instead ?
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Paper 3, Section II, B
2015 commentConsider the dynamical system
where is to be regarded as a fixed real constant and as a real parameter.
Find the fixed points of the system and determine the stability of the system linearized about the fixed points. Hence identify the values of at given where bifurcations occur.
Describe informally the concepts of centre manifold theory and apply it to analyse the bifurcations that occur in the above system with . In particular, for each bifurcation derive an equation for the dynamics on the extended centre manifold and hence classify the bifurcation.
What can you say, without further detailed calculation, about the case ?
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Paper 2, Section II, B
2015 comment(a) An autonomous dynamical system in has a periodic orbit with period . The linearized evolution of a small perturbation is given by . Obtain the differential equation and initial condition satisfied by the matrix .
Define the Floquet multipliers of the orbit. Explain why one of the multipliers is always unity and give a brief argument to show that the other is given by
(b) Use the energy-balance method for nearly Hamiltonian systems to find leading-order approximations to the two limit cycles of the equation
where .
Determine the stability of each limit cycle, giving reasoning where necessary.
[You may assume that and .]
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Paper 1, Section II, 28B
2015 comment(a) What is a Lyapunov function?
Consider the dynamical system for given by
Prove that the origin is asymptotically stable (quoting carefully any standard results that you use).
Show that the domain of attraction of the origin includes the region where the maximum possible value of is to be determined.
Show also that there is a region such that implies that increases without bound. Explain your reasoning carefully. Find the smallest possible value of .
(b) Now consider the dynamical system
Prove that this system has a periodic solution (again, quoting carefully any standard results that you use).
Demonstrate that this periodic solution is unique.
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Paper 4, Section II, A
2015 commentA point particle of charge has trajectory in Minkowski space, where is its proper time. The resulting electromagnetic field is given by the Liénard-Wiechert 4-potential
Write down the condition that determines the point on the trajectory of the particle for a given value of . Express this condition in terms of components, setting and , and define the retarded time .
Suppose that the 3 -velocity of the particle is small in size compared to , and suppose also that . Working to leading order in and to first order in , show that
Now assume that can be replaced by in the expressions for and above. Calculate the electric and magnetic fields to leading order in and hence show that the Poynting vector is (in this approximation)
If the charge is performing simple harmonic motion , where is a unit vector and , find the total energy radiated during one period of oscillation.
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Paper 3, Section II, 34A
2015 comment(i) Consider the action
where is a 4-vector potential, is the field strength tensor, is a conserved current, and is a constant. Derive the field equation
For the action describes standard electromagnetism. Show that in this case the theory is invariant under gauge transformations of the field , which you should define. Is the theory invariant under these same gauge transformations when ?
Show that when the field equation above implies
Under what circumstances does hold in the case ?
(ii) Now suppose that and obeying reduce to static 3 -vectors and in some inertial frame. Show that there is a solution
for a suitable Green's function with as . Determine for any . [Hint: You may find it helpful to consider first the case and then the case , using the result , where
If is zero outside some bounded region, comment on the effect of the value of on the behaviour of when is large. [No further detailed calculations are required.]
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Paper 1, Section II, A
2015 commentBriefly explain how to interpret the components of the relativistic stress-energy tensor in terms of the density and flux of energy and momentum in some inertial frame.
(i) The stress-energy tensor of the electromagnetic field is
where is the field strength, is the Minkowski metric, and is the permeability of free space. Show that , where is the current 4-vector.
[ Maxwell's equations are and ]
(ii) A fluid consists of point particles of rest mass and charge . The fluid can be regarded as a continuum, with 4 -velocity depending on the position in spacetime. For each there is an inertial frame in which the fluid particles at are at rest. By considering components in , show that the fluid has a current 4-vector field
and a stress-energy tensor
where is the proper number density of particles (the number of particles per unit spatial volume in in a small region around ). Write down the Lorentz 4-force on a fluid particle at . By considering the resulting 4 -acceleration of the fluid, show that the total stress-energy tensor is conserved, i.e.
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Paper 4, Section II, E
2015 commentA stationary inviscid fluid of thickness and density is located below a free surface at and above a deep layer of inviscid fluid of the same density in flowing with uniform velocity in the direction. The base velocity profile is thus
while the free surface at is maintained flat by gravity.
By considering small perturbations of the vortex sheet at of the form , calculate the dispersion relationship between and in the irrotational limit. By explicitly deriving that
deduce that the vortex sheet is unstable at all wavelengths. Show that the growth rates of the unstable modes are approximately when and when .
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Paper 2, Section II, E
2015 commentConsider an infinite rigid cylinder of radius a parallel to a horizontal rigid stationary surface. Let be the direction along the surface perpendicular to the cylinder axis, the direction normal to the surface (the surface is at ) and the direction along the axis of the cylinder. The cylinder moves with constant velocity . The minimum separation between the cylinder and the surface is denoted by .
(i) What are the conditions for the flow in the thin gap between the cylinder and the surface to be described by the lubrication equations? State carefully the relevant length scale in the direction.
(ii) Without doing any calculation, explain carefully why, in the lubrication limit, the net fluid force acting on the stationary surface at has no component in the direction.
(iii) Using the lubrication approximation, calculate the component of the velocity field in the gap between the cylinder and the surface, and determine the pressure gradient as a function of the gap thickness .
(iv) Compute the tangential component of the force, , acting on the bottom surface per unit length in the direction.
[You may quote the following integrals:
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Paper 3, Section II, E
2015 commentConsider a three-dimensional high-Reynolds number jet without swirl induced by a force imposed at the origin in a fluid at rest. The velocity in the jet, described using cylindrical coordinates , is assumed to remain steady and axisymmetric, and described by a boundary layer analysis.
(i) Explain briefly why the flow in the jet can be described by the boundary layer equations
(ii) Show that the momentum flux in the jet, , where is an infinite surface perpendicular to , is not a function of . Combining this result with scalings from the boundary layer equations, derive the scalings for the unknown width and typical velocity of the jet as functions of and the other parameters of the problem .
(iii) Solving for the flow using a self-similar Stokes streamfunction
show that satisfies the differential equation
What boundary conditions should be applied to this equation? Give physical reasons for them.
[Hint: In cylindrical coordinates for axisymmetric incompressible flow you are given the incompressibility condition as
the -component of the Navier-Stokes equation as
and the relationship between the Stokes streamfunction, , and the velocity components as
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Paper 1, Section II, E
2015 comment(i) In a Newtonian fluid, the deviatoric stress tensor is linearly related to the velocity gradient so that the total stress tensor is
Show that for an incompressible isotropic fluid with a symmetric stress tensor we necessarily have
where is a constant which we call the dynamic viscosity and is the symmetric part of .
(ii) Consider Stokes flow due to the translation of a rigid sphere of radius so that the sphere exerts a force on the fluid. At distances much larger than the radius of the sphere, the instantaneous velocity and pressure fields are
where is measured with respect to an origin located at the centre of the sphere, and .
Consider a sphere of radius instantaneously concentric with . By explicitly computing the tractions and integrating them, show that the force exerted by the fluid located in on is constant and independent of , and evaluate it.
(iii) Explain why the Stokes equations in the absence of body forces can be written
Show that by integrating this equation in the fluid volume located instantaneously between and , you can recover the result in (ii) directly.
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Paper 4, Section I, B
2015 commentExplain how the Papperitz symbol
represents a differential equation with certain properties. [You need not write down the differential equation explicitly.]
The hypergeometric function is defined to be the solution of the equation given by the Papperitz symbol

that is analytic at and such that . Show that
indicating clearly any general results for manipulating Papperitz symbols that you use.
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Paper 3, Section , B
2015 commentDefine what is meant by the Cauchy principal value in the particular case
where the constant is real and strictly positive. Evaluate this expression explicitly, stating clearly any standard results involving contour integrals that you use.
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Paper 2, Section I, B
2015 commentGive a brief description of what is meant by analytic continuation.
The dilogarithm function is defined by
Let
where is a contour that runs from the origin to the point . Show that provides an analytic continuation of and describe its domain of definition in the complex plane, given a suitable branch cut.
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Paper 1, Section , B
2015 commentEvaluate the real integral
where is taken to be the positive square root.
What is the value of
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Paper 2, Section II, B
2015 commentThe Riemann zeta function is defined by the sum
which converges for . Show that
The analytic continuation of is given by the Hankel contour integral
Verify that this agrees with the integral above when Re and is not an integer. [You may assume .] What happens when ?
Evaluate . Show that is an odd function of and hence, or otherwise, show that for any positive integer .
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Paper 1, Section II, B
2015 commentConsider the differential equation
where and are constants with and . Laplace's method for finding solutions involves writing
for some suitable contour and some suitable function . Determine for the equation and use a clearly labelled diagram to specify contours giving two independent solutions when is real in each of the cases and .
Now let and . Find explicit expressions for two independent solutions to . Find, in addition, a solution with .
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Paper 3, Section II, F
2015 commentLet be of degree , with no repeated roots, and let be a splitting field for .
(i) Show that is irreducible if and only if for any there is such that .
(ii) Explain how to define an injective homomorphism . Find an example in which the image of is the subgroup of generated by (2 3). Find another example in which is an isomorphism onto .
(iii) Let and assume is irreducible. Find a chain of subgroups of that shows it is a solvable group. [You may quote without proof any theorems from the course, provided you state them clearly.]
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Paper 4, Section II,
2015 comment(i) Prove that a finite solvable extension of fields of characteristic zero is a radical extension.
(ii) Let be variables, , and where are the elementary symmetric polynomials in the variables . Is there an element such that but ? Justify your answer.
(iii) Find an example of a field extension of degree two such that for any . Give an example of a field which has no extension containing an primitive root of unity.
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Paper 2, Section II, F
2015 comment(i) State the fundamental theorem of Galois theory, without proof. Let be a splitting field of . Show that is Galois and that Gal has a subgroup which is not normal.
(ii) Let be the 8 th cyclotomic polynomial and denote its image in again by . Show that is not irreducible in .
(iii) Let and be coprime natural numbers, and let and where . Show that .
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Paper 1, Section II,
2015 comment(i) Let be a field extension and be irreducible of positive degree. Prove the theorem which states that there is a correspondence
(ii) Let be a field and . What is a splitting field for ? What does it mean to say is separable? Show that every is separable if is a finite field.
(iii) The primitive element theorem states that if is a finite separable field extension, then for some . Give the proof of this theorem assuming is infinite.
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Paper 4, Section II, D
2015 commentIn static spherically symmetric coordinates, the metric for de Sitter space is given by
where and is a constant.
(a) Let for . Use the coordinates to show that the surface is non-singular. Is a space-time singularity?
(b) Show that the vector field is null.
(c) Show that the radial null geodesics must obey either
Which of these families of geodesics is outgoing
Sketch these geodesics in the plane for , where the -axis is horizontal and lines of constant are inclined at to the horizontal.
(d) Show, by giving an explicit example, that an observer moving on a timelike geodesic starting at can cross the surface within a finite proper time.
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Paper 2, Section II, D
2015 comment(a) The Schwarzschild metric is
(in units for which the speed of light ). Show that a timelike geodesic in the equatorial plane obeys
where
and and are constants.
(b) For a circular orbit of radius , show that
Given that the orbit is stable, show that .
(c) Alice lives on a small planet that is in a stable circular orbit of radius around a (non-rotating) black hole of radius . Bob lives on a spacecraft in deep space far from the black hole and at rest relative to it. Bob is ageing times faster than Alice. Find an expression for in terms of and and show that .
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Paper 3, Section II, D
2015 commentLet be the Levi-Civita connection and the Riemann tensor corresponding to a metric , and let be the Levi-Civita connection and the Riemann tensor corresponding to a metric . Let .
(a) Show that is a tensor.
(b) Using local inertial coordinates for the metric , or otherwise, show that
holds in all coordinate systems, where the semi-colon denotes covariant differentiation using the connection . [You may assume that .]
(c) In the case that for some vector field , show that if and only if
(d) Using the result that if and only if for some scalar field , show that the condition on in part (c) can be written as
for a certain covector field , which you should define.
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Paper 1, Section II, 35D
2015 commentA vector field is said to be a conformal Killing vector field of the metric if
for some scalar field . It is a Killing vector field if .
(a) Show that is equivalent to
(b) Show that if is a conformal Killing vector field of the metric , then is a Killing vector field of the metric , where is any function that obeys
(c) Use part (b) to find an example of a metric with coordinates and (where for which are the contravariant components of a Killing vector field. [Hint: You may wish to start by considering what happens in Minkowski space.]
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Paper 1, Section II, I
2015 comment(a) What does it mean to say that a graph is strongly regular with parameters
(b) Let be an incomplete, strongly regular graph with parameters and of order . Suppose . Show that the numbers
are integers.
(c) Suppose now that is an incomplete, strongly regular graph with parameters . Show that .
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Paper 2, Section II, I
2015 comment(a) Define the Ramsey numbers and for integers . Show that exists for all and that if then .
(b) Show that, as , we have and .
(c) Show that, as , we have and .
[Hint: For the lower bound in (c), you may wish to begin by modifying a random graph to show that for all and we have
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Paper 3, Section II, I
2015 comment(a) Let be a graph. What is a Hamilton cycle in ? What does it mean to say that is Hamiltonian?
(b) Let be a graph of order satisfying . Show that is Hamiltonian. For each , exhibit a non-Hamiltonian graph of order with .
(c) Let be a bipartite graph with vertices in each class satisfying . Show that is Hamiltonian. For each , exhibit a non-Hamiltonian bipartite graph with vertices in each class and .
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Paper 4, Section II, I
2015 commentLet be a bipartite graph with vertex classes and . What does it mean to say that contains a matching from to ? State and prove Hall's Marriage Theorem.
Suppose now that every has , and that if and with then . Show that contains a matching from to .
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Paper 1, Section II, D
2015 commentLet be an evolution equation for the function . Assume and all its derivatives decay rapidly as . What does it mean to say that the evolution equation for can be written in Hamiltonian form?
The modified KdV (mKdV) equation for is
Show that small amplitude solutions to this equation are dispersive.
Demonstrate that the mKdV equation can be written in Hamiltonian form and define the associated Poisson bracket ,} on the space of functionals of u. Verify that the Poisson bracket is linear in each argument and anti-symmetric.
Show that a functional is a first integral of the mKdV equation if and only if , where is the Hamiltonian.
Show that if satisfies the mKdV equation then
Using this equation, show that the functional
Poisson-commutes with the Hamiltonian.
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Paper 2, Section II, D
2015 comment(a) Explain how a vector field
generates a 1-parameter group of transformations in terms of the solution to an appropriate differential equation. [You may assume the solution to the relevant equation exists and is unique.]
(b) Suppose now that . Define what is meant by a Lie-point symmetry of the ordinary differential equation
(c) Prove that every homogeneous, linear ordinary differential equation for admits a Lie-point symmetry generated by the vector field
By introducing new coordinates
which satisfy and , show that every differential equation of the form
can be reduced to a first-order differential equation for an appropriate function.
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Paper 3, Section II, D
2015 commentLet and be real matrices, with symmetric and antisymmetric. Suppose that
Show that all eigenvalues of the matrix are -independent. Deduce that the coefficients of the polynomial
are first integrals of the system.
What does it mean for a -dimensional Hamiltonian system to be integrable? Consider the Toda system with coordinates obeying
where here and throughout the subscripts are to be determined modulo 3 so that and . Show that
is a Hamiltonian for the Toda system.
Set and . Show that
Is this coordinate transformation canonical?
By considering the matrices
or otherwise, compute three independent first integrals of the Toda system. [Proof of independence is not required.]
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Paper 3, Section II, G
2015 commentState and prove the Baire Category Theorem. [Choose any version you like.]
An isometry from a metric space to another metric space is a function such that for all . Prove that there exists no isometry from the Euclidean plane to the Banach space of sequences converging to 0 . [Hint: Assume is an isometry. For and let denote the coordinate of . Consider the sets consisting of all pairs with and .]
Show that for each there is a linear isometry .
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Paper 4, Section II, G
2015 commentLet be a Hilbert space and . Define what is meant by an adjoint of and prove that it exists, it is linear and bounded, and that it is unique. [You may use the Riesz Representation Theorem without proof.]
What does it mean to say that is a normal operator? Give an example of a bounded linear map on that is not normal.
Show that is normal if and only if for all .
Prove that if is normal, then , that is, that every element of the spectrum of is an approximate eigenvalue of .
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Paper 2, Section II, G
2015 comment(a) Let be a linear map between normed spaces. What does it mean to say that is bounded? Show that is bounded if and only if is continuous. Define the operator norm of and show that the set of all bounded, linear maps from to is a normed space in the operator norm.
(b) For each of the following linear maps , determine if is bounded. When is bounded, compute its operator norm and establish whether is compact. Justify your answers. Here for and for .
(i) .
(ii) .
(iii) .
(iv) , where is a given element of . [Hint: Consider first the case that for every , and apply to a suitable function. In the general case apply to a suitable sequence of functions.]
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Paper 1, Section II, G
2015 comment(a) Let be an orthonormal basis of an inner product space . Show that for all there is a unique sequence of scalars such that .
Assume now that is a Hilbert space and that is another orthonormal basis of . Prove that there is a unique bounded linear map such that for all . Prove that this map is unitary.
(b) Let with . Show that no subspace of is isomorphic to . [Hint: Apply the generalized parallelogram law to suitable vectors.]
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Paper 4, Section II, I
2015 commentState the Axiom of Foundation and the Principle of -Induction, and show that they are equivalent (in the presence of the other axioms of ). [You may assume the existence of transitive closures.]
Explain briefly how the Principle of -Induction implies that every set is a member of some .
Find the ranks of the following sets:
(i) ,
(ii) the Cartesian product ,
(iii) the set of all functions from to .
[You may assume standard properties of rank.]
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Paper 3, Section II, I
2015 comment(i) State and prove Zorn's Lemma. [You may assume Hartogs' Lemma.] Where in your proof have you made use of the Axiom of Choice?
(ii) Let be a partial ordering on a set . Prove carefully that may be extended to a total ordering of .
What does it mean to say that is well-founded?
If has an extension that is a well-ordering, must be well-founded? If is well-founded, must every total ordering extending it be a well-ordering? Justify your answers.
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Paper 2, Section II, I
2015 comment(a) Give the inductive and synthetic definitions of ordinal addition, and prove that they are equivalent. Give the inductive definitions of ordinal multiplication and ordinal exponentiation.
(b) Answer, with brief justification, the following:
(i) For ordinals and with , must we have ? Must we have ?
(ii) For ordinals and with , must we have ?
(iii) Is there an ordinal such that ?
(iv) Show that . Is the least ordinal such that ?
[You may use standard facts about ordinal arithmetic.]
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Paper 1, Section II, I
2015 commentState and prove the Completeness Theorem for Propositional Logic.
[You do not need to give definitions of the various terms involved. You may assume the Deduction Theorem, provided that you state it precisely.]
State the Compactness Theorem and the Decidability Theorem, and deduce them from the Completeness Theorem.
Let consist of the propositions for . Does prove ? Justify your answer. [Here are primitive propositions.]
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Paper 4, Section I, E
2015 comment(i) A variant of the classic logistic population model is given by the HutchinsonWright equation
where . Determine the condition on (in terms of ) for the constant solution to be stable.
(ii) Another variant of the logistic model is given by the equation
where . Give a brief interpretation of what this model represents.
Determine the condition on (in terms of ) for the constant solution to be stable in this model.
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Paper 3, Section I, E
2015 commentThe number of a certain type of annual plant in year is given by . Each plant produces seeds that year and then dies before the next year. The proportion of seeds that germinate to produce a new plant the next year is given by where . Explain briefly why the system can be described by
Give conditions on for a stable positive equilibrium of the plant population size to be possible.
Winters become milder and now a proportion of all plants survive each year . Assume that plants produce seeds as before while they are alive. Show that a wider range of now gives a stable positive equilibrium population.
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Paper 2, Section I, E
2015 commentAn activator-inhibitor system is described by the equations
where .
Find the range of for which the spatially homogeneous system has a stable equilibrium solution with and .
For the case when the homogeneous system is stable, consider spatial perturbations proportional to to the equilibrium solution found above. Show that the system has a Turing instability when
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Paper 1, Section I, E
2015 commentThe population density of individuals of age at time satisfies
where is the age-dependent death rate and is the birth rate per individual of age . Show that this may be solved with a similarity solution of the form if the growth rate satisfies where
Suppose now that the birth rate is given by with and is a positive integer, and the death rate is constant in age (i.e. . Find the average number of offspring per individual.
Find the similarity solution, and find the threshold for the birth parameter so that corresponds to a growing population.
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Paper 4, Section II, E
2015 commentIn a stochastic model of multiple populations, is the probability that the population sizes are given by the vector at time . The jump rate is the probability per unit time that the population sizes jump from to . Under suitable assumptions, the system may be approximated by the multivariate Fokker-Planck equation (with summation convention)
where and matrix elements .
(a) Use the multivariate Fokker-Planck equation to show that
[You may assume that as .]
(b) For small fluctuations, you may assume that the vector may be approximated by a linear function in and the matrix may be treated as constant, i.e. and (where and are constants). Show that at steady state the covariances satisfy
(c) A lab-controlled insect population consists of larvae and adults. Larvae are added to the system at rate . Larvae each mature at rate per capita. Adults die at rate per capita. Give the vector and matrix for this model. Show that at steady state
(d) Find the variance of each population size near steady state, and show that the covariance between the populations is zero.
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Paper 3, Section II, E
2015 commentA fungal disease is introduced into an isolated population of frogs. Without disease, the normalised population size would obey the logistic equation , where the dot denotes differentiation with respect to time. The disease causes death at rate and there is no recovery. The disease transmission rate is and, in addition, offspring of infected frogs are infected from birth.
(a) Briefly explain why the population sizes and of uninfected and infected frogs respectively now satisfy
(b) The population starts at the disease-free population size and a small number of infected frogs are introduced. Show that the disease will successfully invade if and only if .
(c) By finding all the equilibria in and considering their stability, find the long-term outcome for the frog population. State the relationships between and that distinguish different final populations.
(d) Plot the long-term steady total population size as a function of for fixed , and note that an intermediate mortality rate is actually the most harmful. Explain why this is the case.
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Paper 4, Section II, H
2015 commentLet be a number field. State Dirichlet's unit theorem, defining all the terms you use, and what it implies for a quadratic field , where is a square-free integer.
Find a fundamental unit of .
Find all integral solutions of the equation .
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Paper 2, Section II, H
2015 comment(i) Let or . Show that remains prime in if and only if is irreducible .
(ii) Factorise , (3) in , when . Compute the class group of .
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Paper 1, Section II,
2015 comment(a) Let be a number field, and a monic polynomial whose coefficients are in . Let be a field containing and . Show that if , then is an algebraic integer.
Hence conclude that if is monic, with , then .
(b) Compute an integral basis for when the minimum polynomial of is .
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Paper 4, Section I, H
2015 commentShow that if is prime then must be a power of 2 . Now assuming is a power of 2 , show that if is a prime factor of then .
Explain the method of Fermat factorization, and use it to factor .
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Paper 3, Section I, H
2015 commentWhat does it mean to say that a positive definite binary quadratic form is reduced? Find the three smallest positive integers properly represented by each of the forms and . Show that every odd integer represented by some positive definite binary quadratic form with discriminant is represented by at least one of the forms and .
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Paper 2, Section I, H
2015 commentDefine the Euler totient function and the Möbius function . Suppose and are functions defined on the natural numbers satisfying . State and prove a formula for in terms of . Find a relationship between and .
Define the Riemann zeta function . Find a Dirichlet series for valid for .
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Paper 1, Section I, H
2015 commentDefine the Legendre symbol . State and prove Euler's criterion, assuming if you wish the existence of primitive roots .
By considering the prime factors of for an odd integer, prove that there are infinitely many primes with .
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Paper 4, Section II, H
2015 commentState the Chinese Remainder Theorem.
Let be an odd positive integer. Define the Jacobi symbol . Which of the following statements are true, and which are false? Give a proof or counterexample as appropriate.
(i) If then the congruence is soluble.
(ii) If is not a square then .
(iii) If is composite then there exists an integer a coprime to with
(iv) If is composite then there exists an integer coprime to with
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Paper 3, Section II, H
2015 commentLet be a real number with continued fraction expansion . Define the convergents (by means of recurrence relations) and show that for we have
Show that
and deduce that as .
By computing a suitable continued fraction expansion, find solutions in positive integers and to each of the equations and .
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Paper 4, Section II, E
2015 comment(a) Define the th Krylov space for and . Letting be the dimension of , prove the following results.
(i) There exists a positive integer such that for and for .
(ii) If , where are eigenvectors of for distinct eigenvalues and all are nonzero, then .
(b) Define the term residual in the conjugate gradient (CG) method for solving a system with symmetric positive definite . Explain (without proof) the connection to Krylov spaces and prove that for any right-hand side the CG method finds an exact solution after at most steps, where is the number of distinct eigenvalues of . [You may use without proof known properties of the iterates of the CG method.]
Define what is meant by preconditioning, and explain two ways in which preconditioning can speed up convergence. Can we choose the preconditioner so that the CG method requires only one step? If yes, is it a reasonable method for speeding up the computation?
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Paper 2, Section II, E
2015 comment(a) The boundary value problem on the unit square with zero boundary conditions and scalar constant is discretised using finite differences as
with . Show that for the resulting system , for a suitable matrix and vectors and , both the Jacobi and Gauss-Seidel methods converge. [You may cite and use known results on the discretised Laplace operator and on the convergence of iterative methods.]
Define the Jacobi method with relaxation parameter . Find the eigenvalues of the iteration matrix for the above problem and show that, in order to ensure convergence for all , the condition is necessary.
[Hint: The eigenvalues of the discretised Laplace operator in two dimensions are for integers .]
(b) Explain the components and steps in a multigrid method for solving the Poisson equation, discretised as . If we use the relaxed Jacobi method within the multigrid method, is it necessary to choose to get fast convergence? Explain why or why not.
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Paper 3, Section II, E
2015 comment(a) Given the finite-difference recurrence
that discretises a Cauchy problem, the amplification factor is defined by
Show how acts on the Fourier transform of . Hence prove that the method is stable if and only if for all .
(b) The two-dimensional diffusion equation
for some scalar constant is discretised with the forward Euler scheme
Using Fourier stability analysis, find the range of values for which the scheme is stable.
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Paper 4, Section II,
2015 commentConsider the scalar system evolving as
where is a white noise sequence with and . It is desired to choose controls to minimize . Show that for the minimal cost is .
Find a constant and a function which solve
Let be the class of those policies for which every obeys the constraint . Show that , for all . Find, and prove optimal, a policy which over all minimizes
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Paper 3, Section II, K
2015 commentA burglar having wealth may retire, or go burgling another night, in either of towns 1 or 2 . If he burgles in town then with probability he will, independently of previous nights, be caught, imprisoned and lose all his wealth. If he is not caught then his wealth increases by 0 or , each with probability and independently of what happens on other nights. Values of and are the same every night. He wishes to maximize his expected wealth at the point he retires, is imprisoned, or nights have elapsed.
Using the dynamic programming equation
with appropriately defined, prove that there exists an optimal policy under which he burgles another night if and only if his wealth is less than .
Suppose and . Prove that he should never burgle in town 2 .
[Hint: Suppose , there are nights to go, and it has been shown that he ought not burgle in town 2 if less than nights remain. For the case , separately consider subcases and . An interchange argument may help.]
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Paper 1, Section II, 38E
2015 comment(a) The diffusion equation
with the initial condition in and zero boundary conditions at and , is solved by the finite-difference method
where , and .
Assuming that the function and the exact solution are sufficiently smooth, prove that the exact solution satisfies the numerical scheme with error for constant .
(b) For the problem in part (a), assume that there exist such that for all . State (without proof) the Gershgorin theorem and prove that the method is stable for .
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Paper 2, Section II,
2015 commentAs a function of policy and initial state , let
where and for all . Suppose that for a specific policy , and all ,
Prove that for all and .
A gambler plays games in which he may bet 1 or 2 pounds, but no more than his present wealth. Suppose he has pounds after games. If he bets pounds then , or , with probabilities and respectively. Gambling terminates at the first such that or . His final reward is . Let be the policy of always betting 1 pound. Given , show that .
Is optimal when ?
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Paper 4, Section II, E
2015 comment(a) Show that the Cauchy problem for satisfying
with initial data , which is a smooth -periodic function of , defines a strongly continuous one parameter semi-group of contractions on the Sobolev space for any .
(b) Solve the Cauchy problem for the equation
with , where are smooth -periodic functions of , and show that the solution is smooth. Prove from first principles that the solution satisfies the property of finite propagation speed.
[In this question all functions are real-valued, and
are the Sobolev spaces of functions which are -periodic in , for
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Paper 3, Section II, E
2015 comment(a) Show that if is a Schwartz function and is a tempered distribution which solves
for some constant , then there exists a number which depends only on , such that for any . Explain briefly why this inequality remains valid if is only assumed to be in .
Show that if is given then for any .
[Hint: The inequality holds for any positive and ]
Prove that if is a smooth bounded function which solves
for some constant vector and constant , then there exists a number such that and depends only on .
[You may use the fact that, for non-negative , the Sobolev space of functions
(b) Let be a smooth real-valued function, which is -periodic in and satisfies the equation
Give a complete proof that if for all then for all and .
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Paper 2, Section II, E
2015 commentProve that if is absolutely integrable with , and for , then for every Schwartz function the convolution
uniformly in as .
Show that the function given by
for satisfies
for . Hence prove that the tempered distribution determined by the function is a fundamental solution of the operator
[You may use the fact that ]
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Paper 1, Section II, E
2015 comment(a) State the Cauchy-Kovalevskaya theorem, and explain for which values of it implies the existence of solutions to the Cauchy problem
where is real analytic. Using the method of characteristics, solve this problem for these values of , and comment on the behaviour of the characteristics as approaches any value where the non-characteristic condition fails.
(b) Consider the Cauchy problem
with initial data and which are -periodic in . Give an example of a sequence of smooth solutions which are also -periodic in whose corresponding initial data and are such that while for non-zero as
Comment on the significance of this in relation to the concept of well-posedness.
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Paper 4, Section II, A
2015 commentThe Hamiltonian for a quantum system in the Schrödinger picture is , where is independent of time and the parameter is small. Define the interaction picture corresponding to this Hamiltonian and derive a time evolution equation for interaction picture states.
Suppose that and are eigenstates of with distinct eigenvalues and , respectively. Show that if the system is in state at time zero then the probability of measuring it to be in state at time is
Let be the Hamiltonian for an isotropic three-dimensional harmonic oscillator of mass and frequency , with being the ground state wavefunction (where ) and being wavefunctions for the states at the first excited energy level . The oscillator is in its ground state at when a perturbation
is applied, with , and is then measured after a very large time has elapsed. Show that to first order in perturbation theory the oscillator will be found in one particular state at the first excited energy level with probability
but that the probability that it will be found in either of the other excited states is zero (to this order).
You may use the fact that
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Paper 3, Section II, A
2015 commentLet denote the normalised joint eigenstates of and , where is the angular momentum operator for a quantum system. State clearly the possible values of the quantum numbers and and write down the corresponding eigenvalues in units with .
Consider two quantum systems with angular momentum states and . The eigenstates corresponding to their combined angular momentum can be written as
where are Clebsch-Gordan coefficients for addition of angular momenta and . What are the possible values of and what is a necessary condition relating and in order that ?
Calculate the values of for and for all . Use the sign convention that when takes its maximum value.
A particle with spin and intrinsic parity is at rest. It decays into two particles and with spin and spin 0 , respectively. Both and have intrinsic parity . The relative orbital angular momentum quantum number for the two particle system is . What are the possible values of for the cases and ?
Suppose particle is prepared in the state before it decays. Calculate the probability for particle to be found in the state , given that .
What is the probability if instead ?
[Units with should be used throughout. You may also use without proof
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Paper 2, Section II, A
2015 commentExpress the spin operator for a particle of spin in terms of the Pauli matrices where
Show that for any unit vector and deduce that
The space of states for a particle of spin has basis states which are eigenstates of with eigenvalues and respectively. If the Hamiltonian for the particle is , find
as linear combinations of the basis states.
The space of states for a system of two spin particles is . Write down explicit expressions for the joint eigenstates of and , where is the sum of the spin operators for the particles.
Suppose that the two-particle system has Hamiltonian and that at time the system is in the state with eigenvalue . Calculate the probability that at time the system will be measured to be in the state with eigenvalue zero.
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Paper 1, Section II, A
2015 commentIf and are operators which each commute with their commutator , show that
By solving this differential equation for , deduce that
The annihilation and creation operators for a harmonic oscillator of mass and frequency are defined by
Write down an expression for the general normalised eigenstate of the oscillator Hamiltonian in terms of the ground state . What is the energy eigenvalue of the state
Suppose the oscillator is now subject to a small perturbation so that it is described by the modified Hamiltonian with . Show that
where is a constant, to be determined. Hence show that to the shift in the ground state energy as a result of the perturbation is
[Standard results of perturbation theory may be quoted without proof.]
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Paper 4, Section II,
2015 commentGiven independent and identically distributed observations with finite mean and variance , explain the notion of a bootstrap sample , and discuss how you can use it to construct a confidence interval for .
Suppose you can operate a random number generator that can simulate independent uniform random variables on . How can you use such a random number generator to simulate a bootstrap sample?
Suppose that and are cumulative probability distribution functions defined on the real line, that as for every , and that is continuous on . Show that, as ,
State (without proof) the theorem about the consistency of the bootstrap of the mean, and use it to give an asymptotic justification of the confidence interval . That is, prove that as where is the joint distribution of
[You may use standard facts of stochastic convergence and the Central Limit Theorem without proof.]
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Paper 3, Section II, J
2015 commentDefine what it means for an estimator of an unknown parameter to be consistent.
Let be a sequence of random real-valued continuous functions defined on such that, as converges to in probability for every , where is non-random. Suppose that for some and every we have
and that has exactly one zero for every . Show that as , and deduce from this that the maximum likelihood estimator (MLE) based on observations from a model is consistent.
Now consider independent observations of bivariate normal random vectors
where and is the identity matrix. Find the MLE of and show that the MLE of equals
Show that is not consistent for estimating . Explain briefly why the MLE fails in this model.
[You may use the Law of Large Numbers without proof.]
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Paper 2, Section II, J
2015 commentConsider a random variable arising from the binomial distribution , . Find the maximum likelihood estimator and the Fisher information for .
Now consider the following priors on :
(i) a uniform prior on ,
(ii) a prior with density proportional to ,
(iii) a prior.
Find the means and modes of the posterior distributions corresponding to the prior distributions (i)-(iii). Which of these posterior decision rules coincide with ? Which one is minimax for quadratic risk? Justify your answers.
[You may use the following properties of the distribution. Its density , is proportional to , its mean is equal to , and its mode is equal to
provided either or .
You may further use the fact that a unique Bayes rule of constant risk is a unique minimax rule for that risk.]
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Paper 1, Section II, J
2015 commentConsider a normally distributed random vector modelled as where is the identity matrix, and where . Define the Stein estimator of .
Prove that dominates the estimator for the risk function induced by quadratic loss
Show however that the worst case risks coincide, that is, show that
[You may use Stein's lemma without proof, provided it is clearly stated.]
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Paper 4, Section II, J
2015 comment(a) State Fatou's lemma.
(b) Let be a random variable on and let be a sequence of random variables on . What does it mean to say that weakly?
State and prove the Central Limit Theorem for i.i.d. real-valued random variables. [You may use auxiliary theorems proved in the course provided these are clearly stated.]
(c) Let be a real-valued random variable with characteristic function . Let be a sequence of real numbers with and . Prove that if we have
then
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Paper 3 , Section II, J
2015 comment(a) Let be a measure space. What does it mean to say that is a measure-preserving transformation? What does it mean to say that a set is invariant under ? Show that the class of invariant sets forms a -algebra.
(b) Take to be with Lebesgue measure on its Borel -algebra. Show that the baker's map defined by
is measure-preserving.
(c) Describe in detail the construction of the canonical model for sequences of independent random variables having a given distribution .
Define the Bernoulli shift map and prove it is a measure-preserving ergodic transformation.
[You may use without proof other results concerning sequences of independent random variables proved in the course, provided you state these clearly.]
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Paper 2, Section II, J
2015 comment(a) Let be a measure space, and let . What does it mean to say that belongs to ?
(b) State Hölder's inequality.
(c) Consider the measure space of the unit interval endowed with Lebesgue measure. Suppose and let .
(i) Show that for all ,
(ii) For , define
Show that for fixed, the function satisfies
where
(iii) Prove that is a continuous function. [Hint: You may find it helpful to split the integral defining into several parts.]
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Paper 1, Section II, J
2015 comment(a) Define the following concepts: a -system, a -system and a -algebra.
(b) State the Dominated Convergence Theorem.
(c) Does the set function
furnish an example of a Borel measure?
(d) Suppose is a measurable function. Let be continuous with . Show that the limit
exists and lies in the interval
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Paper 4, Section II, F
2015 comment(a) Let be the circle group. Assuming any required facts about continuous functions from real analysis, show that every 1-dimensional continuous representation of is of the form
for some .
(b) Let , and let be a continuous representation of on a finitedimensional vector space .
(i) Define the character of , and show that .
(ii) Show that .
(iii) Let be the irreducible 4-dimensional representation of . Decompose into irreducible representations. Hence decompose the exterior square into irreducible representations.
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Paper 3, Section II, F
2015 comment(a) State Mackey's theorem, defining carefully all the terms used in the statement.
(b) Let be a finite group and suppose that acts on the set .
If , we say that the action of on is -transitive if has at least elements and for every pair of -tuples and such that the are distinct elements of and the are distinct elements of , there exists with for every .
(i) Let have at least elements, where and let . Show that acts -transitively on if and only if acts transitively on and the stabiliser acts -transitively on .
(ii) Show that the permutation module can be decomposed as
where is the trivial module and is some -module.
(iii) Assume that , so that . Prove that is irreducible if and only if acts 2-transitively on . In that case show also that is not the trivial representation. [Hint: Pick any orbit of on ; it is isomorphic as a -set to for some subgroup . Consider the induced character
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Paper 2, Section II,
2015 commentLet be a finite group. Suppose that is a finite-dimensional complex representation of dimension . Let be arbitrary.
(i) Define the th symmetric power and the th exterior power and write down their respective dimensions.
Let and let be the eigenvalues of on . What are the eigenvalues of on and on ?
(ii) Let be an indeterminate. For any , define the characteristic polynomial of on by . What is the relationship between the coefficients of and the character of the exterior power?
Find a relation between the character of the symmetric power and the polynomial .
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Paper 1, Section II, F
2015 comment(a) Let be a finite group and let be a representation of . Suppose that there are elements in such that the matrices and do not commute. Use Maschke's theorem to prove that is irreducible.
(b) Let be a positive integer. You are given that the dicyclic group
has order .
(i) Show that if is any th root of unity in , then there is a representation of over which sends
(ii) Find all the irreducible representations of .
(iii) Find the character table of .
[Hint: You may find it helpful to consider the cases odd and even separately.]
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Paper 3, Section II, F
2015 commentLet denote the Weierstrass -function with respect to a lattice and let be an even elliptic function with periods . Prove that there exists a rational function such that . If we write where and are coprime polynomials, find the degree of in terms of the degrees of the polynomials and . Describe all even elliptic functions of degree two. Justify your answers. [You may use standard properties of the Weierstrass -function.]
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Paper 2, Section II, F
2015 commentLet be a domain in . Define the germ of a function element at . Let be the set of all germs of function elements in . Define the topology on . Show it is a topology, and that it is Hausdorff. Define the complex structure on , and show that there is a natural projection map which is an analytic covering map on each connected component of .
Given a complete analytic function on , describe how it determines a connected component of . [You may assume that a function element is an analytic continuation of a function element along a path if and only if there is a lift of to starting at the germ of at and ending at the germ of at .]
In each of the following cases, give an example of a domain in and a complete analytic function such that:
(i) is regular but not bijective;
(ii) is surjective but not regular.
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Paper 1, Section II, F
2015 commentLet be a non-constant holomorphic map between compact connected Riemann surfaces and let denote the set of branch points. Show that the map is a regular covering map.
Given and a closed curve in with initial and final point , explain how this defines a permutation of the (finite) set . Show that the group obtained from all such closed curves is a transitive subgroup of the full symmetric group of the fibre .
Find the group for where .
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Paper 4, Section I, J
2015 commentData on 173 nesting female horseshoe crabs record for each crab its colour as one of 4 factors (simply labelled ), its width (in ) and the presence of male crabs nearby (a 1 indicating presence). The data are collected into the data frame crabs and the first few lines are displayed below.

Describe the model being fitted by the command below.
fit1 <- glm(males colour + width, family = binomial, data=crabs)
The following (abbreviated) output is obtained from the summary command.

Write out the calculation for an approximate confidence interval for the coefficient for width. Describe the calculation you would perform to obtain an estimate of the probability that a female crab of colour 3 and with a width of has males nearby. [You need not actually compute the end points of the confidence interval or the estimate of the probability above, but merely show the calculations that would need to be performed in order to arrive at them.]
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Paper 3, Section I, J
2015 commentData are available on the number of counts (atomic disintegration events that take place within a radiation source) recorded with a Geiger counter at a nuclear plant. The counts were registered at each second over a 30 second period for a short-lived, man-made radioactive compound. The first few rows of the dataset are displayed below.

Describe the model being fitted with the following command.
fit Counts Time, data=geiger)
Below is a plot against time of the residuals from the model fitted above.

Referring to the plot, suggest how the model could be improved, and write out the code for fitting this new model. Briefly describe how one could test in whether the new model is to be preferred over the old model.
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Paper 2, Section I, J
2015 commentLet be independent Poisson random variables with means , where for some known constants and an unknown parameter . Find the log-likelihood for .
By first computing the first and second derivatives of the log-likelihood for , describe the algorithm you would use to find the maximum likelihood estimator . Hint: Recall that if then
for .]
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Paper 1, Section I, J
2015 commentThe outputs of a particular process are positive and are believed to be related to -vectors of covariates according to the following model
In this model are i.i.d. random variables where is known. It is not possible to measure the output directly, but we can detect whether the output is greater than or less than or equal to a certain known value . If
show that a probit regression model can be used for the data .
How can we recover and from the parameters of the probit regression model?
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Paper 4, Section II, J
2015 commentConsider the normal linear model where the -vector of responses satisfies with . Here is an matrix of predictors with full column rank where and is an unknown vector of regression coefficients. For , denote the th column of by , and let be with its th column removed. Suppose where is an -vector of 1 's. Denote the maximum likelihood estimate of by . Write down the formula for involving , the orthogonal projection onto the column space of .
Consider with . By thinking about the orthogonal projection of onto , show that
[You may use standard facts about orthogonal projections including the fact that if and are subspaces of with a subspace of and and denote orthogonal projections onto and respectively, then for all .]
By considering the fitted values , explain why if, for any , a constant is added to each entry in the th column of , then will remain unchanged. Let . Why is (*) also true when all instances of and are replaced by and respectively?
The marks from mid-year statistics and mathematics tests and an end-of-year statistics exam are recorded for 100 secondary school students. The first few lines of the data are given below.

The following abbreviated output is obtained:

What are the hypothesis tests corresponding to the final column of the coefficients table? What is the hypothesis test corresponding to the final line of the output? Interpret the results when testing at the level.
How does the following sample correlation matrix for the data help to explain the relative sizes of some of the -values?

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Paper 1, Section II, J
2015 commentAn experiment is conducted where scientists count the numbers of each of three different strains of fleas that are reproducing in a controlled environment. Varying concentrations of a particular toxin that impairs reproduction are administered to the fleas. The results of the experiment are stored in a data frame in , whose first few rows are given below.

The full dataset has 80 rows. The first column provides the number of fleas, the second provides the concentration of the toxin and the third specifies the strain of the flea as factors 0,1 or 2 . Strain 0 is the common flea and strains 1 and 2 have been genetically modified in a way thought to increase their ability to reproduce in the presence of the toxin.
Explain and interpret the commands and (abbreviated) output below. In particular, you should describe the model being fitted, briefly explain how the standard errors are calculated, and comment on the hypothesis tests being described in the summary.

Explain and motivate the following code in the light of the output above. Briefly explain the differences between the models fitted below, and the model corresponding to it

Denote by the three models being fitted in sequence above. Explain the hypothesis tests comparing the models to each other that can be performed using the output from the following code.
fit1$dev, fit2$dev, fit3$dev)
[1]
[1]
Use these numbers to comment on the most appropriate model for the data.
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Paper 4, Section II, C
2015 commentThe Ising model consists of particles, labelled by , arranged on a -dimensional Euclidean lattice with periodic boundary conditions. Each particle has spin up , or down , and the energy in the presence of a magnetic field is
where is a constant and indicates that the second sum is over each pair of nearest neighbours (every particle has nearest neighbours). Let , where is the temperature.
(i) Express the average spin per particle, , in terms of the canonical partition function .
(ii) Show that in the mean-field approximation
where is a single-particle partition function, is an effective magnetic field which you should find in terms of and , and is a prefactor which you should also evaluate.
(iii) Deduce an equation that determines for general values of and temperature . Without attempting to solve for explicitly, discuss how the behaviour of the system depends on temperature when , deriving an expression for the critical temperature and explaining its significance.
(iv) Comment briefly on whether the results obtained using the mean-field approximation for are consistent with an expression for the free energy of the form
where and are positive constants.
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Paper 3, Section II, C
2015 comment(a) A sample of gas has pressure , volume , temperature and entropy .
(i) Use the first law of thermodynamics to derive the Maxwell relation
(ii) Define the heat capacity at constant pressure and the enthalpy and show that .
(b) Consider a perfectly insulated pipe with a throttle valve, as shown.

Gas initially occupying volume on the left is forced slowly through the valve at constant pressure . A constant pressure is maintained on the right and the final volume occupied by the gas after passing through the valve is .
(i) Show that the enthalpy of the gas is unchanged by this process.
(ii) The Joule-Thomson coefficient is defined to be . Show that
[You may assume the identity ]
(iii) Suppose that the gas obeys an equation of state
where is the number of particles. Calculate to first order in and hence derive a condition on for obtaining a positive Joule-Thomson coefficient.
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Paper 2, Section II, C
2015 comment(a) State the Bose-Einstein distribution formula for the mean occupation numbers of discrete single-particle states with energies in a gas of bosons. Write down expressions for the total particle number and the total energy when the singleparticle states can be treated as continuous, with energies and density of states .
(b) Blackbody radiation at temperature is equivalent to a gas of photons with
where is the volume and is a constant. What value of the chemical potential is required when applying the Bose-Einstein distribution to photons? Show that the heat capacity at constant volume satisfies for some constant , to be determined.
(c) Consider a system of bosonic particles of fixed total number . The particles are trapped in a potential which has ground state energy zero and which gives rise to a density of states , where is a constant. Explain, for this system, what is meant by Bose-Einstein condensation and show that the critical temperature satisfies . If is the number of particles in the ground state, show that for just below
for some constant , to be determined.
(d) Would you expect photons to exhibit Bose-Einstein condensation? Explain your answer very briefly.
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Paper 1, Section II, C
2015 comment(a) Define the canonical partition function for a system with energy levels , where labels states, given that the system is in contact with a heat reservoir at temperature . What is the probability that the system occupies state ? Starting from an expression for the entropy , deduce that
(b) Consider an ensemble consisting of copies of the system in part (a) with very large, so that there are members of the ensemble in state . Starting from an expression for the number of ways in which this can occur, find the entropy of the ensemble and hence re-derive the expression . [You may assume Stirling's formula for large.
(c) Consider a system of non-interacting particles at temperature . Each particle has internal states with energies
Assuming that the internal states are the only relevant degrees of freedom, calculate the total entropy of the system. Find the limiting values of the entropy as and and comment briefly on your answers.
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Paper 4, Section II,
2015 comment(i) An investor in a single-period market with time- 0 wealth may generate any time-1 wealth of the form , where is any element of a vector space of random variables. The investor's objective is to maximize , where is strictly increasing, concave and . Define the utility indifference price of a random variable .
Prove that the map is concave. [You may assume that any supremum is attained.]
(ii) Agent has utility . The agents may buy for time- 0 price a risky asset which will be worth at time 1 , where is random and has density
Assuming zero interest, prove that agent will optimally choose to buy
units of the risky asset at time 0 .
If the asset is in unit net supply, if , and if , prove that the market for the risky asset will clear at price
What happens if
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Paper 3, Section II,
2015 commentA single-period market consists of assets whose prices at time are denoted by , and a riskless bank account bearing interest rate . The value of is given, and . An investor with utility wishes to choose a portfolio of the available assets so as to maximize the expected utility of her wealth at time 1. Find her optimal investment.
What is the market portfolio for this problem? What is the beta of asset ? Derive the Capital Asset Pricing Model, that
Excess return of asset Excess return of market portfolio .
The Sharpe ratio of a portfolio is defined to be the excess return of the portfolio divided by the standard deviation of the portfolio . If is the correlation of the return on asset with the return on the market portfolio, prove that
Sharpe ratio of asset Sharpe ratio of market portfolio .
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Paper 1, Section II,
2015 comment(i) What does it mean to say that is a martingale?
(ii) If is an integrable random variable and , prove that is a martingale. [Standard facts about conditional expectation may be used without proof provided they are clearly stated.] When is it the case that the limit exists almost surely?
(iii) An urn contains initially one red ball and one blue ball. A ball is drawn at random and then returned to the urn with a new ball of the other colour. This process is repeated, adding one ball at each stage to the urn. If the number of red balls after draws and replacements is , and the number of blue balls is , show that is a martingale, where
Does this martingale converge almost surely?
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Paper 2, Section II,
2015 comment(i) What is Brownian motion?
(ii) Suppose that is Brownian motion, and the price at time of a risky asset is given by
where is the constant growth rate, and is the constant volatility of the asset. Assuming that the riskless rate of interest is , derive an expression for the price at time 0 of a European call option with strike and expiry , explaining briefly the basis for your calculation.
(iii) With the same notation, derive the time-0 price of a European option with expiry which at expiry pays
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Paper 4, Section I,
2015 commentLet be the set of all non-empty compact subsets of -dimensional Euclidean space . Define the Hausdorff metric on , and prove that it is a metric.
Let be a sequence in . Show that is also in and that as in the Hausdorff metric.
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Paper 3, Section I,
2015 commentLet be a compact subset of with path-connected complement. If and , show that there is a polynomial such that
for all .
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Paper 2, Section I, I
2015 commentLet be the roots of the Legendre polynomial of degree . Let , be chosen so that
for all polynomials of degree or less. Assuming any results about Legendre polynomials that you need, prove the following results:
(i) for all polynomials of degree or less;
(ii) for all ;
(iii) .
Now consider . Show that
as for all continuous functions .
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Paper 1, Section I, I
2015 commentLet be a non-empty bounded open subset of with closure and boundary . Let be continuous with twice differentiable on .
(i) Why does have a maximum on ?
(ii) If and on , show that has a maximum on .
(iii) If on , show that has a maximum on .
(iv) If on and on , show that on .
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Paper 2, Section II,
2015 commentState and prove Sperner's lemma concerning the colouring of triangles.
Deduce a theorem, to be stated clearly, on retractions to the boundary of a disc.
State Brouwer's fixed point theorem for a disc and sketch a proof of it.
Let be a continuous function such that for some we have for all . Show that is surjective.
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Paper 3, Section II, I
2015 commentLet . By considering the set consisting of those for which there exists an with for all , or otherwise, give a Baire category proof of the existence of continuous functions on such that
at each .
Are the following statements true? Give reasons.
(i) There exists an such that
for each and each .
(ii) There exists an such that
for each and each .
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Paper 4, Section II, 36B
2015 commentThe shallow-water equations
describe one-dimensional flow over a horizontal boundary with depth and velocity , where is the acceleration due to gravity.
Show that the Riemann invariants are constant along characteristics satisfying , where is the linear wave speed and denotes a reference state.
An initially stationary pool of fluid of depth is held between a stationary wall at and a removable barrier at . At the barrier is instantaneously removed allowing the fluid to flow into the region .
For , find and in each of the regions
explaining your argument carefully with a sketch of the characteristics in the plane.
For , show that the solution in region (ii) above continues to hold in the region . Explain why this solution does not hold in
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Paper 2, Section II, 36B
2015 commentA uniform elastic solid with density and Lamé moduli and occupies the region between rigid plane boundaries and . Starting with the linear elastic wave equation, show that SH waves can propagate in the -direction within this waveguide, and find the dispersion relation for the various modes.
State the cut-off frequency for each mode. Find the corresponding phase velocity and group velocity , and sketch these functions for .
Define the time and cross-sectional average appropriate for a mode with frequency energy. [You may assume that the elastic energy per unit volume is .]
An elastic displacement of the form is created in a region near , and then released at . Explain briefly how the amplitude of the resulting disturbance varies with time as at the moving position for each of the cases and . [You may quote without proof any generic results from the method of stationary phase.]
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Paper 3, Section II, B
2015 commentDerive the ray-tracing equations for the quantities and during wave propagation through a slowly varying medium with local dispersion relation , explaining the meaning of the notation .
The dispersion relation for water waves is , where is the water depth, , and and are the components of in the horizontal and directions. Water waves are incident from an ocean occupying onto a beach at . The undisturbed water depth is , where are positive constants and is sufficiently small that the depth can be assumed to be slowly varying. Far from the beach, the waves are planar with frequency and with crests making an acute angle with the shoreline.
Obtain a differential equation (with defined implicitly) for a ray and show that near the shore the ray satisfies
where and should be found. Sketch the shape of the wavecrests near the shoreline for the case .
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Paper 1, Section II, 37B
2015 commentAn acoustic plane wave (not necessarily harmonic) travels at speed in the direction , where , through an inviscid, compressible fluid of unperturbed density . Show that the velocity is proportional to the perturbation pressure , and find . Define the acoustic intensity .
A harmonic acoustic plane wave with wavevector and unitamplitude perturbation pressure is incident from on a thin elastic membrane at unperturbed position . The regions and are both occupied by gas with density and sound speed . The kinematic boundary conditions at the membrane are those appropriate for an inviscid fluid, and the (linearized) dynamic boundary condition
where and are the tension and mass per unit area of the membrane, and (with ) is its perturbed position. Find the amplitudes of the reflected and transmitted pressure perturbations, expressing your answers in terms of the dimensionless parameter
Hence show that the time-averaged energy flux in the -direction is conserved across the membrane.
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Paper 2, Section II, A
2015 commentA right circular cylinder of radius and length has volume and total surface area . Use Lagrange multipliers to do the following:
(a) Show that, for a given total surface area, the maximum volume is
determining the integer in the process.
(b) For a cylinder inscribed in the unit sphere, show that the value of which maximises the area of the cylinder is
determining the integers and as you do so.
(c) Consider the rectangular parallelepiped of largest volume which fits inside a hemisphere of fixed radius. Find the ratio of the parallelepiped's volume to the volume of the hemisphere.
[You need not show that suitable extrema you find are actually maxima.]
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Paper 4, Section II, A
2015 commentDerive the Euler-Lagrange equation for the integral
where is allowed to float, and takes a given value.
Given that is finite, and , find the stationary value of